  Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by
automorphisms. We provide a complete classification of a finite group $G$ in
which every maximal $A$-invariant subgroup containing the normalizer of some
$A$-invariant Sylow subgroup is nilpotent. Moreover, we show that both the
hypothesis that every maximal $A$-invariant subgroup of $G$ containing the
normalizer of some $A$-invariant Sylow subgroup is nilpotent and the hypothesis
that every non-nilpotent maximal $A$-invariant subgroup of $G$ is normal are
equivalent.
