Periodic Shunkov's groups saturated by the direct products of an elementary abelian 2-groups and a simple groups $L_2 (2^m)$

Let $G$ be a periodic Shunkov's group containing an involution. It is proved that if every finite subgroup from $G$ of even order is contained in a subgroup, which is isomorphic to the direct product of an elementary abelian 2-group and a group $L_2 (2^m)$ for some $m \geq 2$, that $G \simeq L_2 (Q) \times V$, where $Q$ is some locally finite field of characteristic 2 and $V$ is a group of period 2.