Periodic groups saturated with finite simple groups $L_4(q)$

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 $L_4(q)$, where $q$ is odd, is isomorphic to $L_4(F)$ for a suitable field $F$ of odd characteristic.