Normality-preserving endomorphism-invariant subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a normality-preserving endomorphism-invariant subgroup if, for every normality-preserving endomorphism $$\sigma$$ of $$G$$, $$\sigma(H)$$ is contained in $$H$$. A normality-preserving endomorphism is an endomorphism with the property that the image of any normal subgroup is normal.

Extreme examples

 * The trivial subgroup is normality-preserving endomorphism-invariant in any group.
 * Every group is normality-preserving endomorphism-invariant in itself.

Examples arising from stronger properties or subgroup-defining functions

 * All fully invariant subgroups, including the derived subgroup (commutator subgroup), as well as members of the derived series and lower central series, are normality-preserving endomorphism-invariant.
 * The Fitting subgroup and solvable radical are both normality-preserving endomorphism-invariant. In fact, they are something stronger: weakly normal-homomorph-containing subgroups.