Periodic groups saturated by finite simple groups

Let $M$ be a set of groups. By definition, a group $G$ is saturated with groups from $M$ (or is saturated with $M$) if every finite subgroup of $G$ is contained in a subgroup which is isomorphic to some element of $M$. The talk is devoted to the problem of classification of periodic groups which are saturated with finite non-abelian simple groups.