Normality-preserving endomorphism-invariance is finite direct power-closed

Statement
Suppose $$H$$ is a normality-preserving endomorphism-invariant subgroup of a group $$G$$, i.e., for any normality-preserving endomorphism $$\alpha$$ of $$G$$, we have $$\alpha(H) \subseteq H$$. Then, for any natural number $$n$$, the subgroup $$H^n$$ is a normality-preserving endomorphism-invariant subgroup in the $$n$$-fold external direct product $$G^n$$

Similar facts

 * Full invariance is finite direct power-closed
 * Bound-word property is finite direct power-closed
 * Normal-homomorph-containment is finite direct power-closed

Opposite facts

 * Characteristicity is not finite direct power-closed

Corollaries

 * Normality-preserving endomorphism-invariant implies finite direct power-closed characteristic