Did the Holocaust occur due to G-d's active involvement, or was it due to Him merely removing His protection from them?
According to Rabbi Yissachar Shlomo Teichtal, author of Eim Habanim Smeicha, Hashem personally orchestrated it. For those unfamiliar with the book, he wrote it while hiding in Budapest, with all sources quoted from memory. He breaks from typical Chareidi thought and encourages aliyah. I will quote from the English translation, as that's the copy i own.
It's a very long book, and i don't know where exactly the quote can be found, but someone said that it's written that people asked him where Hashem is, and he said in the Nazi ovens, i.e. He is the cause of this.
In any case, simply looking at some of the subtitles show that this was indeed his opinion.
In chapter 1 IV, The sole purpose of our afflictions is to arouse us to return to the Holy Land.
In chapter 3 I, in the section titled Today's hardships are God's way of telling us to return to our Land, he writes:
Now, however, the Gentiles have taken away our very right to live; they have deprived us of all means of sustenance; they have broken our staff of bread. Therefore, it is clear that the Holy One Blessed be He has removed this "promise of exile." It is as if He is telling us explicitly, "My children, I do not want you to remain in the lands of exile anymore. Therefore, I will no longer protect your stay in Diaspora. Rise up, go to your mother's bosom, and return to the Land of your forefathers." (emphasis mine)
Also, in section VI, there is a part titled Hashem afflicts us so that we will return to our Land, which again says that Hashem is doing this to cause us to return to Israel.
So, according to some, Hashem was certainly personally actively involved in bringing out the Holocaust.
I might think that not all would agree on this matter.
However, the psalmist seems very convinced that Jewish defeat and slaughter are very much in the hand of God. My Hebrew skills are not sufficient to say whether or not he suggests that "for your sake we are being killed" indicates that it is the reason that they are being targeted by their enemies or that they die as a kind of sacrifice to God. See verses 20-25 below:
JPS Psalm 44: 1For the Leader; [a Psalm] of the sons of Korah. Maschil. 2O God, we have heard with our ears, our fathers have told us; A work Thou didst in their days, in the days of old. 3Thou with Thy hand didst drive out the nations, and didst plant them in; Thou didst break the peoples, and didst spread them abroad. 4For not by their own sword did they get the land in possession, Neither did their own arm save them; But Thy right hand, and Thine arm, and the light of Thy countenance, Because Thou wast favourable unto them. 5Thou art my King, O God; Command the salvation of Jacob. 6Through Thee do we push down our adversaries; Through Thy name do we tread them under that rise up against us. 7For I trust not in my bow, Neither can my sword save me. 8But Thou hast saved us from our adversaries, And hast put them to shame that hate us. 9In God have we gloried all the day, And we will give thanks unto Thy name for ever. Selah 10Yet Thou hast cast off, and brought us to confusion; And goest not forth with our hosts. 11Thou makest us to turn back from the adversary; And they that hate us spoil at their will. 12Thou hast given us like sheep to be eaten; And hast scattered us among the nations. 13Thou sellest Thy people for small gain, And hast not set their prices high. 14Thou makest us a taunt to our neighbours, A scorn and a derision to them that are round about us. 15Thou makest us a byword among the nations, A shaking of the head among the peoples. 16All the day is my confusion before me, And the shame of my face hath covered me, 17For the voice of him that taunteth and blasphemeth; By reason of the enemy and the revengeful. 18All this is come upon us; yet have we not forgotten Thee, Neither have we been false to Thy covenant. 19Our heart is not turned back, Neither have our steps declined from Thy path; 20Though Thou hast crushed us into a place of jackals, And covered us with the shadow of death. 21If we had forgotten the name of our God, Or spread forth our hands to a strange god; 22Would not God search this out? For He knoweth the secrets of the heart. 23Nay, but for Thy sake are we killed all the day; We are accounted as sheep for the slaughter. 24Awake, why sleepest Thou, O Lord? Arouse Thyself, cast not off for ever. 25Wherefore hidest Thou Thy face, And forgettest our affliction and our oppression? 26For our soul is bowed down to the dust; Our belly cleaveth unto the earth. 27Arise for our help, And redeem us for Thy mercy’s sake. THE HOLY SCRIPTURES ACCORDING TO THE MASORETIC TEXT A NEW TRANSLATION Jewish Publication Society 1917
As pointed out in a comment, the appeal to God to "wake up" suggests some ambiguity.
The Ohr Hachayim Hakadosh mentions the oft quoted verse of Eicha, From above does not come forth the bad. This full verse is actually quoting the unbeliever as saying that both the bad and the good does not come from above. However, it is read this way as a Drush, very often.
The Medrash uses this verse to mean that good and evil is brought about by the person himself.
The Rambam in Moreh Nevuchim also describes the punishment as losing Hashgacha.
In summary, what happens is definitely within God's plan and permission. But it is in the form of the elements of the world playing out their true colors. Today, as well, there are plenty of those who would gladly commit another holocaust. Thank God they aren't getting their way. When we stopped meriting God's protection the outside elements do their thing. God did intervene at the moment it was to end.
The Gemara in Megilla mentions a related concept, that God is not happy when the evil are punished but He causes others to be happy. This generates an effect of suffering while others standby and laugh, that the whole universe is against you. But God Himself is not actually rejoicing.
Daniel's prayer in Daniel 9 suggests that the Holocaust was due to your first alternative:
“Lord, the great and awesome God, who keeps his covenant of love with those who love him and keep his commandments, 5 we have sinned and done wrong. We have been wicked and have rebelled; we have turned away from your commands and laws. 6 We have not listened to your servants the prophets, who spoke in your name to our kings, our princes and our ancestors, and to all the people of the land.
7 “Lord, you are righteous, but this day we are covered with shame—the people of Judah and the inhabitants of Jerusalem and all Israel, both near and far, in all the countries where you have scattered us because of our unfaithfulness to you. 8 We and our kings, our princes and our ancestors are covered with shame, Lord, because we have sinned against you. 9 The Lord our God is merciful and forgiving, even though we have rebelled against him; 10 we have not obeyed the Lord our God or kept the laws he gave us through his servants the prophets. 11 All Israel has transgressed your law and turned away, refusing to obey you.
“Therefore the curses and sworn judgments written in the Law of Moses, the servant of God, have been poured out on us, because we have sinned against you. 12You have fulfilled the words spoken against us and against our rulers by bringing on us great disaster. Under the whole heaven nothing has ever been done like what has been done to Jerusalem. 13 Just as it is written in the Law of Moses, all this disaster has come on us, yet we have not sought the favor of the Lord our God by turning from our sins and giving attention to your truth. 14 The Lord did not hesitate to bring the disaster on us, for the Lord our God is righteous in everything he does; yet we have not obeyed him.
15 “Now, Lord our God, who brought your people out of Egypt with a mighty hand and who made for yourself a name that endures to this day, we have sinned, we have done wrong. 16 Lord, in keeping with all your righteous acts, turn away your anger and your wrath from Jerusalem, your city, your holy hill. Our sins and the iniquities of our ancestors have made Jerusalem and your people an object of scorn to all those around us.