Normality-preserving endomorphism-invariant implies characteristic

Statement with symbols
Suppose $$H$$ is a normality-preserving endomorphism-invariant subgroup of a group $$G$$, i.e., for any fact about::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) uses::Automorphism implies normality-preserving endomorphism

Proof
The proof follows directly from fact (1).