On a finite group in which every non-nilpotent maximal invariant subgroup of order divisible by $p$ is normal

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. For any fixed prime divisor $p$ of $\mathrm{\mid }G\mathrm{\mid }$, we prove that if every non-nilpotent maximal $A$-invariant subgroup of $G$ of order divisible by $p$ is normal then $G$ has a Sylow tower and $G$ is $q$-nilpotent or $q$-closed for each prime divisor $q$ of $\mathrm{\mid }G\mathrm{\mid }$.