Analogue of focal subgroup theorem for Hall subgroups

Statement

Suppose $G$ is a finite group and $H$ is a Hall subgroup (?) of $G$. The analogue of the focal subgroup theorem states the following: Let $\operatorname {Foc} _{G}(H)$ denote the focal subgroup of $H$ in $G$:

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

Then, we have:

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