Finite groups in which some particular non-nilpotent maximal invariant subgroups have indices a prime-power

Abstract

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms, assume that $G$ has a maximal $A$-invariant subgroup $M$ that is a direct product of some isomorphic simple groups, we prove that if $G$ has a non-trivial $A$-invariant normal subgroup $N$ such that $N\le M$ and every non-nilpotent maximal $A$-invariant subgroup $K$ of $G$ not containing $N$ has index a prime-power and the projective special linear group $PS{L}_2(7)$ is not a composition factor of $G$, then $G$ is solvable.