Interestingly, the Nazis did NOT consider Karaites they encountered in the Russian Caucasus to be Jews. If you're not following the Oral Torah, you're not a Jew, you're a minim. And what separates Jews from every one else is the belief God gave Moses a separate set of teachings to explain what the Written Torah meant.
You are a little confused. The Nazis killed Eastern European Rabbinic and Karaite Jews. The confusion in Crimea, where two primarily non-Israelite Jewish communities, Karaylar and Krymchaks lived. These Turkic Jews are descended from Khazar and Kipchak converts as well as other Jews. The Karaylar are Crimean Karaites. They divided t hemselves from Jews under the Czars claiming to be Turks of the Mosaic faith. The Nazis continued this. However, the Nazis did consider the Krimchaks (Turkic Rabbinate Jews) to be Jews. Many Karaylar joined Muslim Crimean Tatars in turning over the Krimchaks. After the war, Stalin deported them all. There are virtually no Krymchaks left.
Israel considers all Karaites Jews.
Rabbinic Jews are Jews. The others aren't. Orthodox Jews would not consider Karaites Jews, since they don't accept the divine nature of the Oral Torah. I'd put Karaites in the same category as Samaritans, who by the way, are not Jews either.
(Denny Crane: "Sometimes you can only look for answers from God and failing that... and Fox News".)