Homomorphism extension problem for subdirect products of finite groups

Motivated by the simplification of decomposition formulas for fibred bisets, we study the homomorphism extension problem for subdirect products of finite groups when the codomain is an abelian group satisfying certain hypothesis. We prove that every homomorphism of subdirect products whose Goursat quotients have trivial Schur multipliers is extensible. We also examine the case where subdirect products contain a twisted diagonal subgroup and investigate the functoriality of the extensibility property.