Normality-preserving endomorphism-balanced subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a normality-preserving endomorphism-balanced subgroup of $$G$$ if, for every normality-preserving endomorphism $$\sigma$$ of $$G$$, the restriction of $$\sigma$$ to $$H$$ is a normality-preserving endomorphism of $$H$$.