On periodic groups saturated with a finite set of groups

Let $G$ be a periodic group saturated with a finite set of groups of the form $L\times E$, where $E$ is a finite elementary abelian 2-group and $L$ is a finite simple non-abelian group, which is not isomorphic to $E_6(q)$, $^2E_6(q)$, $U_n(q)$ or $L_n(q)$ for odd $q$ and $n\geq 4$. We prove that $G$ is finite.