Automorph-approximate left transiter of normal is p-Sylow-automorphism-invariant in finite p-groups

Property-theoretic statement
The automorph-approximate left transiter of the property of being a normal subgroup of group of prime power order is the property of being a p-Sylow-automorphism-invariant subgroup of finite p-group (modulo the condition that the whole group is a finite $$p$$-group).

Statement with symbols
Let $$p$$ be a prime number. Let $$G$$ be a finite $$p$$-group, i.e., a group of prime power order where the underlying prime is $$p$$. The following are equivalent for a subgroup $$H$$ of $$G$$:


 * 1) $$H$$ is a p-Sylow-automorphism-invariant subgroup of $$G$$.
 * 2) For any finite $$p$$-group $$K$$ containing $$G$$ as a normal subgroup, there is an automorphism $$\sigma$$ of $$H$$ such that $$\sigma(H)$$ is a normal subgroup of $$K$$.

Related facts

 * Left transiter of normal is characteristic
 * Left transiter of normal is p-automorphism-invariant in p-groups