Normal-potentially characteristic implies normal-extensible automorphism-invariant

Verbal statement
Any normal-potentially characteristic subgroup of a group is a normal-extensible automorphism-invariant subgroup.

Normal-potentially characteristic subgroup
A subgroup $$H \le G$$ is normal-potentially characteristic if there exists a group $$K$$ containing $$G$$ as a normal subgroup such that $$H$$ is a characteristic subgroup of $$K$$.

Normal-extensible automorphism-invariant subgroup
An automorphism $$\sigma$$ of a group $$G$$ is termed a normal-extensible automorphism if, whenever $$K$$ is a group containing $$G$$ as a normal subgroup, there exists an automorphism $$\sigma'$$ of $$K$$ whose restriction to $$G$$ equals $$\sigma$$. A normal-extensible automorphism-invariant subgroup is a subgroup invariant under all normal-extensible automorphisms.

Intermediate properties

 * Normal-potentially relatively characteristic subgroup:

Proof
Given: $$H \le G$$. There exists a group $$K$$ such that $$G$$ is normal in $$K$$ and $$H$$ is characteristic in $$K$$. $$\sigma$$ is a normal-extensible automorphism of $$G$$.

To prove: $$\sigma(H) = H$$.

Proof: Since $$\sigma$$ is normal-extensible, there exists $$\sigma' \in \operatorname{Aut}(K)$$ such that the restriction of $$\sigma'$$ to $$G$$ equals $$\sigma$$. Since $$H$$ is characteristic in $$K$$, $$\sigma'(H) = H$$, and hence, $$\sigma(H) = H$$.