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:

$H$ is a Fully normalized subgroup (?) of $G$ , i.e., every automorphism of $H$ extends to an inner automorphism of $G$ .
$H$ 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 $C_G(H)$ , $\sigma(H) \le H$ .