Maybe there is a new movement of people who like to call themselves Karaites. They can make up any believe system they want. Karaites, who still exist and have a small community in Israel and there are some from Egypt, are called Karaites, not Jews. Is there some reason why you WANT to believe they are actually Jews? Does this serve some purpose for you? You aren't Jewish, and you don't understand Jewish law, so how do you offer yourself up as an expert on the definition of Judaism? According to your logic, Jews are actually Christians, even if they don't believe in Christianity, because they have a similar foundation in the Torah (five books of Moses.) Frankly, I don't get why this is even being debated. Judaism dismissed the practice of Karaitism as not being Judaism a long time ago. As we have seen, even today, there are people who call themselves Jews who are actually not Jewish. If you would like to insist that Karaites are Jews, that is your deal. If you would like to find some Jews on FR who would like to join you in your belief, please go ahead. To say that you are still practicing Judaism if you don't accept Oral Law is nonsense.
In 4543 (733 AD), there lived in Beth-Zur, a town not far from Hebron, a man by the name of 'Anan ענן a scholar of the then Gaon, Rabbi Yehudai, of Babylon. He had observed in his scholar that he had neither affection for, nor faith in our tradition as an exposition of the written law; wherefore he ('Anan) could not be chosen either as Resh Gelutha or Gaon. He returned, therefore, to his native country, Palestine, and formed a new sect, the leader of which he became, by openly preaching against our system of tradition. The sect of the Sadducees, who only adopted the written law and rejected the tradition, had gradually fallen entirely into decay after the destruction of the temple and Jerusalem, and had become nearly dissolved. But the appearance of 'Anan gave them new life, and they soon increased and spread extensively in Palestine, Egypt, and North Africa. In Palestine, they had yet another learned chief, Sheich Abu al Ferag, who wrote a work, bearing his own name, concerning the principles of his sect, and which contains much that is absurd and blasphemous. He is the same Abu al Ferag who is frequently mentioned in the Opinions of Maimonides תשובות הרמב״ם. Anan had a wife who was called Al Me'alma, i. e. the learned, the instructress, who was acknowledged chief of his sect after the death of her husband, and was consulted in all cases of doubt; and as everything was decided according to her opinion and practice, it came to be customary to ask among the sect, "How did Al Me'alma on that occasion? or what was her practice in that case?" and every one looked up to her for guidance. When, at a later period, Rabbi Joseph Ben Ali became Nahssi in Africa, he used every effort to suppress this sect in all directions,--so that it was nearly dissolved, and but few vestiges are found thereof at the present time; since all that is left are the few Caraites קראים who only acknowledge the written law, are partly descended from the ancient Sadducees, and are found in several places in Asia and Egypt. We find mentioned in Abn Ezra's Commentary on the Pentateuch, several ridiculous expositions of many passages of Holy Writ ascribed to a certain Anan; it would, therefore, appear that he also had composed a commentary on the books of Moses.
From: History of Palestine
Wrong, Karaites call themselves Jews, and view Karaism as a sect (the original sect, in their view) of Judaism. That's what they've always believed, and what you'll see on any Karaite website or pamphlet.
Years ago I did read several books related to Judaism, including fragments of English translation of Talmud (I did not have time or patience to read everything). One of my readings was Faith Strengthened by Isaac from Troki written in about 1500AD. It is probably the best Jewish polemics against Christianity and was given to me by one rabbi.
Also Karaites are related to the Polish history and I knew a Polish Karaite from old family. This made me interested in their history.
Another reason was the debate I had with Protestant friends about canon of the Bible. Protestants rejected Christian biblical tradition of the first 15 centuries and instead they embraced the Masoretic anti-Karaite version from 800AD.
According to your logic, Jews are actually Christians, even if they don't believe in Christianity, because they have a similar foundation in the Torah (five books of Moses.)
You are not far of the mark. I do think that the Church and the Synagogue are the two branches of the same religion. The key difference is what is believed to be the Redemption/Tikkun. Those Jews who became the first Christians accepted the teaching of Jesus that the Kingdom of God is not of this world and that man has higher heavenly calling.
Those Jews who thought that Messiah/Christ (Christos is a Greek word for the Anointed One) will repair this world, will bring freedom and justice to the Jews and to whole world decided to wait for the other one. (Still they expected Messiah to come at that time and it was one of the motives for the mass uprisings against Roman oppressors).
In the post-Christian Jewish/Talmudic perspective the man is created primarily for this world and this world has to be brought to order.
Before the split the both interpretations - the spiritualist/otherworldly one and second more down to the earth were present among Jews and got separated after Jesus. I assume that this moment of fatal decision is referred by the warning of Saint John the Baptist: "Repent, for the Kingdom of the Heavens is now close at hand"