Groups saturated by finite simple groups

We describe periodic groups saturated by groups in the set a" = {L (2)(r), r a (c) 3/4 4; L (3)(2 (m) ), m a (c) 3/4 1; U (3)(2 (m) ), m a (c) 3/4 2; Sz(2 (m) ), m a (c) 3/4 3}. 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.