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

Definition
Suppose $$p$$ is a prime number and $$P$$ is a finite $$p$$-group, so $$P$$ is a group of prime power order. A subgroup $$H$$ of $$P$$ is termed a $$p$$-core-automorphism-invariant subgroup if $$H$$ satisfies the following equivalent conditions:


 * 1) $$H$$ is invariant under $$O_p(\operatorname{Aut}(P))$$, i.e., the p-core of the automorphism group of $$P$$.
 * 2) Every normal $$p$$-subgroup of $$\operatorname{Aut}(P)$$ sends $$H$$ to itself.