DY
I first heard about modern Karaties in a book about The founding of Israel. I had heard of them of course in school but thought they disappeared.
In the book it talked about the founding of the state of Israel and how many jews from Sephardic countries wanted to emigrate there. The Karaties came from Sephardic countries and also wanted to emigrate. And the book also discussed other Safek jews like the Bnei Israel and later the Ethiopians. With the law of return the issue of who is a jew is not only a halachic one , but a political one as only jews can immigrate to the state of israel.
The Book said that rabbis at the time decided that Karaties where jewish but the issue of marriage , divorce and manzer was an issue and never answered it (It never mentioned Patralineal decent).
For those who say some dont consider them jewish, there are those who dont consider many AMERICAN jews jewish, especially those that came before WW II. In order to marry in Israel you need to prove you are jewish. Most american jews who came before WW II cannot prove this to easily. Think about it, How do you prove you are jewish, Other than you were told you are, you dont have any other physical proof. Even proving your maternal grandmothers maiden last name sounding jewish hasnt been enough proof (Thats about the best proof most people have)
At least the Russian jews had either a Jewish Star or Zhid stamped in their passports so they had some proof.