Quotient-isomorph-containing subgroup

Definition
Suppose $$G$$ is a group and $$H$$ is a subgroup. We say that $$H$$ is a quotient-isomorph-containing subgroup of $$G$$ if the following is true: $$H$$ is a normal subgroup of $$G$$, and if $$K$$ is a normal subgroup of $$G$$ such that the quotient groups $$G/H$$ and $$G/K$$ are isomorphic groups, then $$H \le K$$.

If $$G$$ is a finite group, this property is equivalent to being a quotient-isomorph-free subgroup.

Related properties

 * Isomorph-containing subgroup is a subgroup that contains every isomorphic subgroup to it in the whole group.