On a periodic Shunkov group saturated by direct products of finite elementary abelian 2-groups and $L_2(2^n)$

Let $\Re$ be a set of groups. A group $G$ is said to be saturated by groups from $\Re$ if any finite subgroup from $G$ is contained in a subgroup of $G$ isomorphic to some group from $\Re$. It is proved that a periodic Shunkov group saturated by groups from the set $\Re=\{L_2(2^k)\times I_n\mid n\in N\}$, where $I_n$ is the direct product of $n$ copies of groups of order 2 and $k$ is a fixed number, is locally finite.