Groups embeddable as normal subgroups in a finite group with a common complement

Definition
Suppose $$G$$ and $$H$$ are finite groups. We say that $$G$$ and $$H$$ are embeddable sd normal subgroups in a finite group with a common complement if there exists a finite group $$C$$ with a defining ingredient::normal subgroup $$A$$ isomorphic to $$G$$, a normal subgroup $$B$$ isomorphic to $$H$$, and a subgroup $$D$$ such that $$A,D$$ are permutable complements and $$B,D$$ are defining ingredient::permutable complements.

Equivalently $$D$$ is a retract of $$C$$ having both $$A$$ and $$B$$ as defining ingredient::normal complements.

Weaker relations

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

Facts

 * If we remove the condition of normality on both embeddings, then the only constraint we get is that the two groups have the same order. This follows from the fact that any finite group of order $$n$$ is a permutable complement to the symmetric group on $$n-1$$ letters in the symmetric group on $$n$$ letters by Cayley's theorem.
 * If we impose the additional constraint that the permutable complement itself be a normal subgroup, then the two groups are forced to be isomorphic.