Subgroup for which the transfer to the quotient by its focal subgroup is surjective

Definition
Suppose $$G$$ is a group and $$H$$ is a subgroup of finite index in $$G$$. We say that $$H$$ is a subgroup for which the transfer to the quotient by its focal subgroup is surjective' if the transfer homomorphism $$G \to H/\operatorname{Foc}_G(H)$$ is a surjective homomorphism.

Stronger properties

 * Weaker than::Sylow subgroup:
 * Weaker than::Hall subgroup: