Groups embeddable as normal subgroups in a finite group with isomorphic quotient groups

Definition
Suppose $$G$$ and $$H$$ are finite groups. We say that $$G$$ and $$H$$ are embeddable as normal subgroups in a finite group with isomorphic quotient groups if there exists a group $$A$$ with normal subgroups $$B,C$$ such that:


 * $$B$$ is isomorphic to $$G$$.
 * $$C$$ is isomorphic to $$H$$.
 * $$A/B$$ is isomorphic to $$A/C$$.

Stronger relations

 * Weaker than::Groups embeddable as normal subgroups in a finite group with a common complement
 * Weaker than::Groups embeddable as normal subgroups in a finite group with isomorphic complements

Weaker relations

 * Stronger than::Composition factor-equivalent groups: