I'd like to preface by saying I believe that the positions (1-5) laid out are a little overly simplistic and don't necessarily capture the minutia of these groups, and what we know about them.
I have done my best to lay out a more comprehensive overview of each group, relying upon sources, so the reader can develop a more encapsulating view of each group's position on the Oral Torah. I also want to state that since the Mishna was codified well before the end of the Ancient Era, It would be difficult for any group to believe that the Oral law is a modern, or medieval invention. (The Ancient Era ended c. 500 CE, the medival period stretches from then until c. 1500 CE, and the Modern Era begins c. 1500 CE).
Lastly, I'm going to add this position to the pool:
#6: G-d gave the Oral Torah that’s presented by Orthodox Judaism, and it cannot be changed.
But to simply answer the question:
Sadducces: #2, but per the Rambam, they could be #5 or #6 in secret.
(The Rambam says that the Saddokim made up their position and arguments in order to avoid keeping the Torah because they didn't think they would receive reward, see below.)
Karaites: #2
Samaritans: #2
Beta-Israel: Most likely, and in accordance with their traditions, #6, with the caveat that they did not have access to the Rabbinic teachings due to persecution and diaspora and were not in line with Orthodoxy until the late modern era.
Conservative Judaism: #6 or #5 originally, leaning more towards #5 or #4 nowadays.
Reform Judaism: #2
Saddokim/Sadducces
It's important to remember the Saddokim existed before the codification of the Mishnah, that being said, they did reject many of the teachings of the Perushim (Pharisees), who are the progenitors of "Orothodox Judaism." There were certain positions where the Saddokim were more strict than the Perushim, and some where they were less strict. A very important difference was a disagreement on the laws of Niddah (M. Niddah 4.2) They also seemingly did not hold by any eruv at all (M. Eruvin 6a, see The Mishna with Obadiah Bartenura zl by Rabbi Shraga Silverstein zl). Other disputes include whether or not one who provides false testimony is put to death even if his testimony never actually lead to an execution, (Makkot 5b.8). They also had multiple disputes over temple practices (Yoma 53a.2 ; M. Yadayim 4 ; Chagigah 23a).
Saddokim generally were literalist in interpretation and relied heavily upon explicit references in the Torah, similar to later Karaites (Sanhedrin 33b.11 ; Rambam zl Hilchot Bi'at HaMikdash 1.3 ; see also Meg. Tannit, Nisan where they interpret "an eye for an eye..." literally.)
The Rambam zl says that all of the Sadducean ideology was a farce in order to not follow the Torah, and that these were not their genuine beliefs at all, just a ploy (Rambam on Pirkei Avot 1.3).
The 1st century historian Josephus says "the Pharisees have delivered to the people a great many observances by succession from their fathers, which are not written in the laws of Moses; and for that reason it is that the Sadducees reject them, and say that we are to esteem those observances to be obligatory which are in the written word, but are not to observe what are derived from the tradition of our forefathers" (Antiquities of the Jews, 13.10.6).
He also describes them as not believing in reward and punishment in the World to Come, as well as other seriously divergent beliefs (The War of the Jews 2:8). However, Josephus is not always entirely reliable as a historian even within his own time period, and this work was written c. 95 CE, about 250 years after the Hashmonean victory (161-141 BCE).
Yehuda Ha-Levi zl and Rashi zl also state that the minim discussed in Berachot 54a.8 and elsewhere, who do not believe in The World to Come, are Saddokim, and the Tanchuma has a similar teaching (Al-Kuzari 3.65 ; Rashi on Berakhot 54a.8.2 ; Midrash Tanchuma, Bereshit 5), but this is contested elsewhere, see footnote 261 here (click on the number "261" to exapand). It is explained that (according to this view) this is a misinterpretation caused by non-jewish religious censors who frequently censored sections of the Talmud.
The Boethusim, who are a closely related derevation from the Saddokim, attempt to fool the Perushim and cause them to err in declaring the new moon (Rosh Hashana 22b.7) Kaufmann Kohler says that "In the latter the Sadducees are replaced by the late Boethusians, who had, only for the sake of opposition, maintained certain Sadducean traditions without a proper understanding of the historical principles upon which they were based"(Jewish Encyclopedia 1906, Sadducees) hence the actions of the Boethusim might not be indicidive of Sadducean beliefs.
Even though our understanding of the Saddokim is fragmented, it is very clear they did not accept the Pharisaic tradition that would go on to become Orthodox Judaism.
Karaim/Karaites
Similar in many ways to the Saddokim, The Karaites are referred to as Saddokim by Ibn Ezra zl (Ibn Ezra on Leviticus 11:26:1
) and the Rambam zl (Rambam on Pirkei Avot 1.3). While there is no actual connection through the ages between the two groups (See the position attributed to Rav Sadia Gaon zl, pg 6), they are referred to this way because they have similar ideologies, in that they are biblical maximalists who do not not hold by the Pharisaic/Rabbinic positions de jure.
The Karaite positions overwhelmingly agrees with the Pharisaic position when there is a debate between the Perushim and Saddokim. (Rabbi Revel zl, The Karaite Halakah, 16-51)
While Yehuda Ha-Levi calls the Saddokim minim (as mentioned above), he contrasts this by saying the Karaites are well-informed on "Toldoth," and are attempting to understand the roots of the faith, and are possibly just misled by their foolishness (Al-Kuzari 3.65). It would seem that even according to a the harsher view of the Saddokim, the Karaites are not on as low of a level.
The Karaites were studious and were at times quoted by Rabbinic Jewish leaders for their Biblical textual analysis and understanding of Hebrew grammar (Ibn Ezra on Genesis 28:12), but they were lacking in their understanding, as Ibn Ezra says, "Each one interprets verses as he sees fit. They do the same even with respect to commandments and laws. They are ignorant of the form of Hebrew and therefore err even in grammar. How can people rely on their opinion, with regard to the precepts when they are always changing their mind, moving from one extreme to the other?"
(Ibn Ezra zl on Genesis, Introduction, 16).
Rabbi Revel zl tells us that "the laws on which all Karaites agree are few" (Rabbi Revel, The Karaite Halakah, 3) but the general position of Karaites is similar to this explanation of their position laid out by Rabbi Revel, "As stated in [Kiddushin 66a], the disagreement between John Hyrcanus and the teachers of the Law resulted in the extermination of the latter, excepting Simeon b. Shatah. As a consequence, ignorance of the Law prevailed until Simeon appeared and reinstated it. Simeon, say the Karaites, being at that time the sole authority, introduced many innovations upon his return and changed the true interpretation of the Law. To enforce these new laws, he invented the [say the Karaites,] fiction that besides the Written there is also an Oral Law given to Moses on Sinai and handed down from generation to generation, and that the laws proclaimed by him went back to this real tradition.
The people followed him blindly. But some of them, knowing the false basis of these changes, rejected them and adhered to the ancient Tradition in all its purity; those were the Karaites." (Ibid, 5)
If we recognize this as the position of the Karaites, Then they entirely deny the existence of an Oral Torah. The Shulchan Aruch states they are not Mumarim (ie. they are not apostates/forming a new religion) (Shulchan Arukh, Yoreh De'ah 159).
Shomerim/Samaritans/Kutim
The Samaritans believe that Mt Gerizim, not Mt Moriah (ie. Jerusalem), is the "dwelling place of God's glory," (Memar Marqah, 2.10) and "Israel's place of worship."(Memar Marqah, 2.12) They believe Mt. Gerizim is the correct and original location of the temple (Kitab al-Tariq).
Abu'l-Fath ibn Abi al-Hasan, the Samaritan, in his compilation of earlier Samaritan writings, said (summarized by me): When Phineas zl was the high priest, Eli [the High Priest zl] sought to usurp the priesthood, eventually leading him to splinter off and head for Shiloh [my comment- compare to Onias IV, this story is possibly derivative]. Abu'l Fath goes on to state that Shmuel [Ha-Navi zl] was taken to Eli when he was a baby, where he was taught witchcraft. Shortly thereafter, these Jews (the non-Samaritans) decided upon Mt Moriah as the location for the temple. He says there was not one prophet among the Jews who spoke the truth and that they relied upon sorcery and astrology. He later says Ezra and Zerrubabel conspired and rewrote the Torah to fit their position. (from Kitab al Tariq, available in English translations: Paul Stenhouse, The Kitab al-Tarikh of Abu 'l-Fath or; Tradition Kept: The Literature of the Samaritans) Abu'l Fath's position is that the original Jews kept solely the Torah Scroll, nothing more (Ibid, Excerpt 6).
John MacDonald says that the Samaritans do not borrow ideas from mainstream Judaism post-Ezra. (Theology of the Samaritans, 452.) However they do take much from Islam and the followers of yeshu. (Ibid, 454-456 ; Tradition Kept: The Literature of the Samaritans, 274-277) aswell as Greek thought (Theology of the Samaritans, 30, 456.)
Samaritans have developed separate legal texts in order to guide their performance of the biblical commandments, aswell as liturgies (formalized prayers).
It seems that from the time of Ezra onward, the Samaritans understanding of the oral law developed separately from mainstream Judaism, for example it seems they would not eat a kos-kos (see footnote 28) (Tractate Kutim 1.12). However you can eat meat they butchered, if they are willing to eat it (Kutim 2.1 ; (Chullin 3a.4). This would suggest they do share some traditions, further evidenced by Chullin 4a.10. The discussion in Chullin 4a .9-.17, indicates that concerning the mitzvot that the Samaritans found evidence for in the 5 books of Moses, they kept the additional rules to a sufficient degree (whether that was through their own separate tradition or adoption of the mainstream rabbinic halachot in specific use-cases, is unclear to me. Eitherway, it seems as though they would view this as man-made customs, not Oral Law).
Samaritans do not accept the Rabbinic/mainstream oral tradition, they also do not accept the prophets, the writings, and certain sections from the Masoretic Torah. The Shulchan Aruch renders them as Mumarim (apostates/forming a new religion) (Shulchan Arukh, Yoreh De'ah 159).
Beta-Israel/Ethiopian Jewry
Nowadays many of Beta-Israel follow Rabbinic Judaism, I will speak to Beta-Israel's development before their arrival in Israel.
Beta Israel developed their own practices cut off from Rabbinic Judaism, and seemingly, most of the Oral Torah. For example, they were unaware of the laws of tefillin in the modern era (From Sinai to Ethiopia Daily Pract. 4). According to one of Beta-Israel's explanations, this was because they were forbidden from keeping the mitzvot for some period of time (presumably by their non-Jewish neighbors) and forgot many of the laws (Ibid). This explanation would lead us to believe they had the Oral Torah (and held by it), but forgot it due to persecution.
Conversely, one could say they were unaware of this mitzvah because they were not descended from traditional Rabbinic Judaism (like the karaites who interpret the verse homiletically, or a group that had no knowledge of the idea altogether).
This book goes through Beta-Israel's practices in more detail: From Sinai to Ethiopia.
While it appears as though they didn't follow oral tradition from Sinai (or at least they did not follow very much of it), it is likely that this was due to ignorance rather than rejection (see the fact that they offered sacrifices until a foreigner pointed out to them that this was incorrect (Ibid, Daily Pract. 6.)).
Conservative Judaism
From The Jewish Encyclopedia 1906 on Zecharias Frankel, The man commonly considered to be the progenitor of Conservatism, "[Zecharias Frankel] held firmly the belief that reason based on scholarship and not mere desire on the part of the laity must be the justification for Reform. In this sense Frankel declared himself when the president of the Teplitz congregation expressed the hope that the new rabbi would introduce reforms and do away with the [abuses]. He stated that he knew of no abuses; and that if there were any it was not at all the business of the laity to interfere in such matters [...]. Still he introduced some slight modifications in the worship, as the [removal] of the piyyuṭim, the introduction of a choir of boys, and the like. He was, however, strenuously opposed to any innovation which was objectionable to Jewish sentiment. [He opposed the sentiment that it was permitted for] Jewish high-school boys to write on the Sabbath [...]" (The Jewish Encyclopedia 1906, Zecharias Frankel, Religious Attitude).
It continues,
"His position in the controversy on [a new liberal prayer book] displeased both parties; the Liberals were dissatisfied because, instead of declaring that their prayer-book was in accord with Jewish tradition, he pointed out inconsistencies from the historical and dogmatic points of view; and the Orthodox were dissatisfied because he declared changes in the traditional ritual [theoretically] permissible [...]. A great impression was produced [when] he announced his secession from the rabbinical conference then in session in that city [...], and stated that he could not cooperate with a body of rabbis who had passed a resolution declaring the Hebrew language unnecessary for public worship" (Ibid).
Since its founding Conservatism has crept further towards Reformism, and has also fragmented. While Frankel was purely interested in critical analysis and reforms based on scholarship, modern conservatism seems to be more interested in reforms based on social justice, modern philosophy, and the like.
The current position of the (Conservative) Rabbinical Assembly states:
The mitzvot evolved through a long process of interpretation and debate into the body of law known as halakhah.
Based on the dual pillars of written Torah (the Bible) and
Oral Torah (which includes the Talmud, Codes, Teshuvot—
legal rulings of later rabbis), halakhah allows us to seek
God’s will and apply that will in each generation. Because
the halakhah remains the basis for all authentic Jewish
practice, Conservative Judaism recognizes that no living
body of law can be frozen or fossilized, and holds dear
the notion that the Torah is meant to serve us as a road
“towards” the knowledge of God, not as a barrier keeping
us back. [...] the interpretation and application of these rules is
shaped anew by emerging realities, innovative technologies, and new insights. (Rabbinical Assembly, About Us, Conservative/Masorti Movement, (quote is from "Conservative Judaism: Covenant and Commitment" which is linked there).
Masorti Olami (major Conservative organization) states that "We practice a robust and Halachic Judaism firmly rooted in Torah and Mitzvot but fully integrated into our modern-day lives. We distinguish ourselves by examining the values of egalitarianism, pluralism, humanism, tolerance and democracy in the development of Jewish tradition alongside traditional halachic sources." (Masorti Olami, About Us)
and elsewhere,
"...The authors of both the Written Torah and the Oral Torah were influenced by the social, scientific, legal and, even to a degree, moral context of their times. Although the extraordinary and visionary idea of ethical monotheism transformed how key values and concerns were perceived, and led to the transformation of ancient narratives, compare the ancient precedents with the Torah’s account of The Flood, outlawing human sacrifice, idolatry and numerous forms of immorality and requiring the pursuit of justice and loving kindness, nevertheless the authors of this great work which we call Judaism were limited by the contexts of their times." (Masorti Olami, Masorti Ideology, Religion And Truth).
Stating earlier on the page, "I do not believe that God told us to root out Amalek and the seven nations who inhabited the land previous to the arrival of the Children of Israel there, a course of action which we call ethnic cleansing today. Rather, the authors of that section of the Torah understood God’s role in their history in that way at that time, or had reasons for wanting us to think of it in that manner" (Ibid).
So while the original position of Conservatism might have been within #6, The contemporary position might be closer to #5 (That last quote however, and the rest of its source text, ventures well into #4, even denying a section of the Torah, if I understand his position correctly). Of course these are just the views of their respective organizations, other organizations and individuals that identify as Conservative range from more to less traditional.
Neil Gillman, a philosophy professor at the RA’s Jewish Theology Seminary, urged Conservative Judaism to “abandon its claim that we are a halachic movement,” which he called “irrelevant to the vast majority of our lay people” (JTA Article).
A few of the reforms of the Rabbinical Assembly can be surveyed here
Reform Judaism/Progressive Judaism
The Jewish Encyclopedia 1906 on Reform Judaism, this section being written by Reform Jews, says the following: "Talmudic legalism certainly was a product of the Talmudic period. It was not originally inherent in Judaism. It must not be accepted as eternally obligatory upon later generations. But was Biblical law, perhaps, the original, divinely established norm and form of Judaism, and, as such, binding upon all subsequent generations? If it was, then Reform Judaism, ignoring post-Biblical development and tradition, was identical with Karaism; [ie. If this were the case, Reform Judaism would still have to find a way to follow all biblical laws, merely rejecting orthodox interpretation] and, furthermore, [Reform Judaism's] omission of all reference to [priestly] and sacrificial institutions, though these form an integral part of the Mosaic law and revelation, is in violation of the assumption that Judaism is Law, which Law divinely revealed is the Pentateuch."
continuing,
"This was the dilemma with which Reform theologians were confronted[...]. To meet it, a distinction was drawn between the moral and the ceremonial laws, though certainly the Torah nowhere indicates such distinction nor discloses or fixes the criteria by which the difference is to be established" (Ibid).
The entry implies that Reform Judaism attempted to jettison "ritual" law, however, it says, this did not include "the Sabbath and the other Biblical holy days, circumcision, and in certain circles the dietary laws" (Ibid).
Modern Reform Judaism is a highly liberal theology that emphasizes the "ever-changing nature of Judaism", and attempts to attenuate Jewish personhood with modern progressive values.
The Movement for Reform Judaism says,
"Torah min HaShamayim [The idea of Torah coming from Heaven] is deeply flawed [...] we see that [the Torah] is not a unity but a collection of documents woven together [...] we see that the authorship must be in question[...]. We believe that the Torah [The Written Torah] is a human creation – written by our ancestors and inspired by their understanding of themselves and the place of God in their lives – so ‘divine’ in one sense, but utterly human[...]. We can say that some laws have ‘eternal truth’ while others are a product of their time". (Movement for Reform Judaism, Torah)
"The rabbis utterly transformed a biblical ritual cult into the Judaism that we live today in response to the historical context in which they found themselves" (Movement for Reform Judaism, Authenticity).
"Indeed, none of the Judaism we have today, no matter what our denomination, comes straight from the Bible. It was these rabbinic ancestors who, adapting to crisis and the threat to their culture that came out of their historical reality, gave us the Talmuds, the law codes, and the responsa, all of which are a matrix of law and story "(Ibid, Halachah and Progressive Integrity).
Later,
"Progressive Jewish integrity compels us a Progressive movement as well as individual Progressive Jews to openly reach in to this unending inheritance, these uncountable forms, this matrix of teaching and story, and wrestle elements of it forth into the present. Unafraid to re-shape our prayers, to re-think festivals and the Jewish year, to grasp transforming acts such as conversion and marriage, and to transform them for the present[...]. Progressive Judaism gave priority to ethics; in a conflict with tradition, the ethical choice will determine the direction. Ethics, of course, change. We see this today in a world where, unthinkable 15 years ago, same sex couples marry in our synagogues." (Ibid).
It seems as though the Reform position is that both the Oral and Written Torah are the product of man, and that while they have significance in terms of tradition, Reform Judaism finds modern progressive positions preferable to the Torah, if they are at odds. Once again, these are only the views of these authors/organizations, different individuals within Reform might be more or less in line with tradition.
