Quotient-to-subgroup implications for finite groups

Classify all pairs of finite groups (H_1,H_2) such that every finite group with a quotient isomorphic to H_1 contains a subgroup isomorphic to H_2.

Background

The paper introduces the relation H_1⇒H_2 when every finite extension having quotient H_1 contains a subgroup H_2. For finite abelian groups, the fundamental theorem of finite abelian groups gives a complete answer: the implication holds exactly when H_2 is isomorphic to a subgroup of H_1.

The non-abelian case is largely unresolved in the paper. The question arises naturally from the paper’s use of the fact that sufficiently large elementary abelian subquotients force elementary abelian subgroups, and asks how far analogous quotient-to-subgroup principles extend beyond abelian groups.