Finite groups in which every maximal invariant subgroup of order divisible by $p$ is nilpotent

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 provide a complete characterization of the structure of a group $G$ in which every maximal $A$-invariant subgroup of order divisible by $p$ is nilpotent.