On $$S\Phi$$-Supplemented Subgroups of Finite Groups by Coprime Actions

Let $$G$$ and $$A$$ be finite groups of relative coprime orders and suppose $$A$$ acts on $$G$$ via automorphisms. In this paper, we investigate the $$p$$ -nilpotency and solvability of finite groups in which every maximal $$A$$ -invariant subgroup of a Sylow $$p$$ -subgroup of $$G$$ is an $$S\Phi$$ -supplemented subgroup for some prime $$p$$ .