Subisomorph-containing iff strongly closed in any ambient group

This article gives a proof/explanation of the equivalence of multiple definitions for the term subisomorph-containing subgroup
View a complete list of pages giving proofs of equivalence of definitions

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 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