On finite groups in which some particular maximal invariant subgroups have indices a prime-power

Suppose that A and G are finite groups such that A acts coprimely on G by automorphisms, we first prove that if every maximal A-invariant subgroup of G that contains the normalizer of some A-invariant Sylow subgroup has index a prime-power and the projective special linear group PSL2(7) is not a composition factor of G, then G is solvable. Moreover, we prove that if every non-nilpotent maximal A-invariant subgroup of G has index a prime-power and PSL2(7) is not a composition factor of G, then G is solvable.