They include as special cases CIT-groups where the centralizer of any involution is a 2-group, and TI-groups where any Sylow 2-subgroups have trivial intersection.
The simple C-groups were determined by Suzuki (1965), and his classification is summarized by Gorenstein (1980, 16.4).
The ones whose Sylow 2-subgroups are elementary abelian were classified in a paper of Burnside (1899), which was forgotten for many years until rediscovered by Feit in 1970.
The C-groups include as special cases the TI-groups (trivial intersection groups), that are groups in which any two Sylow 2-subgroups have trivial intersection.
These were classified by Suzuki (1964), and the simple ones are of the form PSL2(q), PSU3(q), Sz(q) for q a power of 2.