Periodic groups saturated with $L_3(2^m)$

Let $\mathfrak M$ be a set of finite groups. A group $G$ is saturated with groups from $\mathfrak M$ if every finite subgroup of $G$ is contained in a subgroup isomorphic to some member of $\mathfrak M$. It is proved that a periodic group $G$ saturated with groups from the set $\{L_3(2^m)\mid m=1,2,\dots\}$ is isomorphic to $L_3(Q)$, for a locally finite field $Q$ of characteristic 2; in particular, it is locally finite.