Subgroup for which the transfer to its abelianization is surjective

Definition
Suppose $$H$$ is a subgroup of finite index in a group $$G$$. We say that $$H$$ is a subgroup for which the transfer to its abelianization is surjective if the transfer homomorphism $$G \to H/[H,H]$$, to the fact about::abelianization of $$H$$, is surjective.