The complete reducibility of some $GF(2)A_7$-modules

It is proved that, if $G$ is a finite group with a nontrivial normal $2$-subgroup $Q$ such that $G/Q\cong A_7$ and an element of order $5$ from $G$ acts without fixed points on $Q$, then the extension of $G$ by $Q$ is splittable, $Q$ is an elementary abelian group, and $Q$ is the direct product of minimal normal subgroups of $G$ each of which is isomorphic, as a $G/Q$-module, to one of the two $4$-dimensional irreducible $GF(2)A_7$-modules that are conjugate with respect to an outer automorphism of the group $A_7$.