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 fact about::fully normalized subgroup of $$G$$, i.e., every automorphism of $$H$$ extends to an inner automorphism of $$G$$.
 * 2) $$H$$ is a fact about::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 fact about::centralizer-annihilating endomorphism-invariant subgroup of $$G$$: For every endomorphism $$\sigma$$ of $$G$$ whose kernel contains $$C_G(H)$$, $$\sigma(H) \le H$$.

Related facts

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