On periodic groups isospectral to $A_7$

The spectrum of a periodic group $G$ is the set $\omega(G)$ of its element orders. Consider a group $G$ such that $\omega(G)=\omega(A_7)$. Assume that $G$ has a subgroup $H$ isomorphic to $A_4$, whose involutions are squares of elements of order $4$. We prove that either $O_2(H) \subseteq O_2(G)$ or $G$ has a finite nonabelian simple subgroup.