On a Shunkov group saturated by central extensions of cyclic groups by projective special linear groups

Let $G$ be a group, and let $\mathfrak R$ be some set of groups. We say that the group $G$ is saturated by groups from the set $\mathfrak R$ if any finite subgroup of $G$ is contained in a subgroup of $G$ isomorphic to some group from $\mathfrak R$. We prove that a periodic Shunkov group saturated by groups from $\mathfrak R=\{L_2(2^n)\times(t_m)\mid n=1,2,\dots,\ m=1,2,\dots,\}$, where $(|L_2(2^n)|,|t_m|)=1$, or from $\mathfrak R=\{L_2(5)\times\langle v\rangle\}$, where $|v|=2^k$, $k=1,2,\dots$, is locally finite.