Groups saturated by finite simple groups

We describe periodic groups saturated by groups in the set
$$ \mathfrak C=\{ L_2(r),\ r\ge4;\quad L_3(2^m),\ m\ge1;\quad U_3(2^m),\ m\ge2;\quad Sz(2^m),\ m\ge3\}. $$
As a corollary we give a description of periodic groups $G$ saturated by finite simple groups and satisfying one of the following conditions: (a) centralizers of involutions in $G$ are 2-closed; (b) $G$ contains a strongly embedded 2-local subgroup.