Subisomorph-containing iff strongly closed in any ambient group

Statement
The following are equivalent for a subgroup $$H$$ of a group $$G$$:


 * $$H$$ is a subisomorph-containing subgroup of $$G$$, i.e., if $$K$$ is a subgroup of $$G$$ isomorphic to a subgroup of $$H$$, then $$K$$ is contained in $$H$$.
 * For any group $$L$$ containing $$G$$, $$H$$ is a fact about::strongly closed subgroup of $$K$$ with respect to $$L$$.

Related facts

 * Isomorph-containing iff weakly closed in any ambient group
 * Same order iff potentially conjugate
 * Isomorphic iff potentially conjugate, isomorphic iff potentially conjugate in finite
 * Inner automorphism to automorphism is right tight for normality
 * Every injective endomorphism arises as the restriction of an inner automorphism