Periodic groups saturated by finite simple groups $U_3(2^m)$

Let $\mathfrak M$ be a set of finite groups. A group $G$ is said to be saturated by the groups in $\mathfrak M$ if every finite subgroup of $G$ is contained in a subgroup isomorphic to a member of $\mathfrak M$. It is proved that a periodic group $G$ saturated by groups in a set $\{U_3(2^m)\mid m=1,2,\dots\}$ is isomorphic to $U_3(Q)$ for some locally finite field $Q$ of characteristic 2; in particular, $G$ is locally finite.