Centralizer-annihilating endomorphism-invariant subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a centralizer-annihilating endomorphism-invariant subgroup of $$G$$ if, for every centralizer-annihilating endomorphism $$\sigma$$ of $$G$$, $$\sigma(H)$$ is contained in $$H$$. Here, a centralizer-annihilating endomorphism of $$G$$ with respect to $$H$$ is an endomorphism whose kernel contains the centralizer $$C_G(H)$$.