Karaimite Jews are a sect of Crimean Jews who began to pretend to be only Karaimites from 1795 in order to gain privileges from Catherine the Great. In the 19th century, their leader Sima Babovich employed a Karaimite called Avraham Firkovich as a teacher to Crimean Jewish children in order to learn how to imitate the ways of the Karaimites.
However, the "Karaimite" Jews maintained a distinction between themselves and the real Karaimites until Seraya Shapshal began to breach the boundaries. Subsequently the "Karaimite" Jews expelled Seraya Shapshal in 1919 because of his friendliness towards real Karaimites.
In 1931 the same "Karaimite" Jews attended the socialist conference in Moscow to eliminate Judaism and during WW2, cashing in again on their history of pretending to be real Karaimites when it suited them, the "Karaimite" Jews under Tuvia Levi-Babovich and Кумыш, Марк Моисеевич (Караман) then colluded with the Nazis in Lutsk, Kiev, Babi Yar even as far as Krasnodar to eradicate the real Karaimites who might have been able to testify against them.
Meanwhile Seraya Shapshal had already been received well by the Karaimite Karaylar in Poland back in 1928 and during WW2 attempted to save as many real Karaimites as he could by identifying them as Crimean Karaites.
After WW2 500 "Karaimite" Jews fled Crimea with the Nazi evacuations.
Under the Soviets, the Turco-Mongol Karaimite-Karaylar having been worn down by religious persecution surrendered themselves to secularization.
In 2008, the "Karaimite" Jewish leader David El tried to restart Seraya Shapshal's efforts in Evpatoria towards the Russian Karaimites and reached out by sending a letter to the World Congress of the European Karaim.
The (by that time fully) Secular Turco-Mongol Karaylar scolded the real Karaimites who welcomed the gesture from David El's "Karaimite" Jews.
These days, modern "Karaimite" Jews now only refer to themselves as Karaite Jews.