Subgroup whose focal subgroup equals its intersection with the derived subgroup

Definition
Suppose $$G$$ is a group and $$H$$ is a subgroup. Denote by $$\operatorname{Foc}_G(H)$$ the focal subgroup of $$H$$ in $$G$$:

$$\operatorname{Foc}_G(H) = \langle xy^{-1} \mid x,y \in H, \exists g \in G, gxg^{-1} = y \rangle$$.

$$H$$ is termed a subgroup whose focal subgroup equals its intersection with the commutator subgroup if we have:

$$\operatorname{Foc}_G(H) = H \cap [G,G]$$.

Note that in general, we only have the containment $$\operatorname{Foc}_G(H) \le H \cap [G,G]$$.

Stronger properties

 * Weaker than::Sylow subgroup:
 * Weaker than::Direct factor
 * Weaker than::Normal subgroup whose focal subgroup equals its intersection with the commutator subgroup