P-Sylow-automorphism-invariant subgroup of finite p-group

Definition
Suppose $$p$$ is a prime number and $$P$$ is a finite $$p$$-group, i.e., a group of prime power order where the underlying prime is $$p$$. A subgroup $$H$$ of $$P$$ is termed a $$p$$-Sylow-automorphism-invariant subgroup of $$P$$ if there exists a $$p$$-Sylow subgroup $$S$$ of $$\operatorname{Aut}(G)$$ such that $$H$$ is invariant under $$S$$.