Theoretically, if (Gd forbid) ISIS took over Israel, and the Jew in question was not immediately killed because he was Jewish and was not in the financial situation to pay the required taxes, should the Jew allow himself to be executed or become a practicing Muslim?
As Mark Twain (supposedly) said, history may not repeat itself, but it sure does rhyme.
Lo and behold, there was Islamic extremism 800 years ago, when Rabbi Moses Maimonides was alive -- and he left a paper trail.
If you were to open up the Yad HaChazaka (his code of law), it would appear that if there is a time when there is a concerted effort to wipe out Judaism, then we would demand a Jew give up his life even for small things -- which would mean he couldn't claim to convert to another faith. (Now ask an interesting question, is a group like ISIS considered "a time of concerted effort", or just a band of individual rogues -- my sense is that if some poor hapless Jewish tourist got lost and found himself in the middle of ISIS-controlled Iraq, he could claim he bumped into some rogues and preserve his own life; whereas if God forbid an extremist Islamic group took over Israel, I'd call that "a time of concerted effort against our faith.") Yet in Maimonides' letters to Jews actually in such situations, of massive forced conversions to Islam (it was in Yemen then, I believe?), he told them to do whatever they needed to stay alive. There are some suggestions on how to resolve this, but at first glance, there may be a theoretical law on the books, but practically we recommend that people stay alive.
Note that the Jews of medieval France and Germany chose death over conversion to Christianity, whereas many from Spain a few centuries later chose to become crypto-Jews. It's been observed that the Ashkenazic vs. Spanish commentaries on the Binding of Isaac actually mirror these differing attitudes towards martrydom.
The rambam wrote responsa concerning this. See igeres hashmad where he clearly says choose islam not death, as apposed to Christianity which is complete heresy, where one must give up their life for kiddush Hashem.