Groups embeddable as normal subgroups in a finite group with isomorphic complements

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 complements if there exists a finite group $$A$$ with normal subgroups $$B,C$$ and subgroups $$D,E$$ such that:


 * $$B$$ is isomorphic to $$G$$.
 * $$B$$ and $$D$$ are permutable complements: in particular, $$A$$ is an internal semidirect product of $$B$$ and $$D$$.
 * $$C$$ is isomorphic to $$H$$.
 * $$C$$ and $$E$$ are permutable complements: in particular, $$A$$ is an internal semidirect product of $$C$$ and $$E$$.
 * $$D$$ is isomorphic to $$E$$.

Stronger relations

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

Weaker relations

 * Stronger than::Groups embeddable as normal subgroups in a finite group with isomorphic quotient groups
 * Stronger than::Composition factor-equivalent groups