P-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 prime is $$p$$). A subgroup $$H$$ of $$P$$ is termed a p-automorphism-invariant subgroup if it satisfies the following equivalent conditions:


 * 1) $$H$$ is invariant under all the $$p$$-automorphisms of $$P$$, where a $$p$$-automorphism is an automorphism whose order is a power of $$p$$.
 * 2) $$H$$ is a subnormal stability automorphism-invariant subgroup of $$P$$.