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

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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 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 permutable complements.

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

Relation with other relations

Weaker relations

Facts