On central Frattini extensions of finite groups

An extension of a group A by a group G is thought of here simply as a group H containing A as a normal subgroup with quotient H/A isomorphic to G. It is called a central Frattini extension if A is contained in the intersection of the centre and the Frattini subgroup of H. The result of the paper is that, given a finite abelian A and finite G, there exists a central Frattini extension of A by G if and only if A can be written as a direct product A = U × V such that U is a homomorphic image of the Schur multiplicator of G and the Frattini quotient of V is a homomorphic image of G.