Normality-preserving endomorphism-invariant implies characteristic

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normality-preserving endomorphism-invariant subgroup) must also satisfy the second subgroup property (i.e., characteristic subgroup)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about normality-preserving endomorphism-invariant subgroup|Get more facts about characteristic subgroup

Statement

Statement with symbols

Suppose $H$ is a normality-preserving endomorphism-invariant subgroup of a group $G$, i.e., for any Normality-preserving endomorphism (?) $\alpha$ of $G$, $\alpha (H)\subseteq H$. Then, $H$ is a characteristic subgroup of $G$, i.e., it is invariant under every automorphism of $G$.

Facts used

1. Automorphism implies normality-preserving endomorphism

Proof

The proof follows directly from fact (1).