Fully normalized and potentially fully invariant implies centralizer-annihilating endomorphism-invariant

Statement

Suppose $H$ is a subgroup of a group $G$ satisfying the following two conditions:

1. $H$ is a Fully normalized subgroup (?) of $G$, i.e., every automorphism of $H$ extends to an inner automorphism of $G$.
2. $H$ is a Potentially fully invariant subgroup (?) of $G$, i.e., there exists a group $K$ containing $G$ in which $H$ is a fully invariant subgroup.

Then, $H$ is a Centralizer-annihilating endomorphism-invariant subgroup (?) of $G$: For every endomorphism $\sigma$ of $G$ whose kernel contains $C_{G}(H)$, $\sigma (H)\leq H$.

Related facts

• Normal not implies potentially fully invariant
• Normal not implies potentially verbal
• Normal not implies image-potentially fully invariant