Periodic Groups Saturated with Finite Simple Groups L4(q)

If M is a set of finite groups, then a group G is said to be saturated with the set M (saturated with groups in M) if every finite subgroup of G is contained in a subgroup isomorphic to some element of M. It is proved that a periodic group with locally finite centralizers of involutions, which is saturated with a set consisting of groups L₄(q), where q is odd, is isomorphic to L₄(F) for a suitable field F of odd characteristic.