Isomorph-containing implies characteristic

Statement
Any isomorph-containing subgroup of a group is a characteristic subgroup.

Isomorph-containing subgroup
A subgroup $$H$$ of a group $$G$$ is termed an isomorph-containing subgroup if, for every subgroup $$K$$ of $$G$$ isomorphic to $$H$$, $$K \le H$$.

Characteristic subgroup
A subgroup $$H$$ of a group $$G$$ is termed a characteristic subgroup if, for every automorphism $$\sigma$$ of $$G$$, $$\sigma(H) \le H$$.

Stronger facts

 * Weaker than::Isomorph-containing implies intermediately characteristic
 * Weaker than::Isomorph-containing implies injective endomorphism-invariant
 * Weaker than::Isomorph-containing implies intermediately injective endomorphism-invariant

Proof
Given: An isomorph-containing subgroup $$H$$ of a group $$G$$.

To prove: For every automorphism $$\sigma$$ of $$G$$, $$\sigma(H) \le H$$

Proof: Since $$\sigma$$ is an automorphism, the restriction of $$\sigma$$ to $$H$$ is an isomorphism from $$H$$ to $$\sigma(H)$$. Hence, $$\sigma(H)$$ is isomorphic to $$H$$. Since $$H$$ is isomorph-containing, we get $$\sigma(H) \le H$$, completing the proof.