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 ${\mathrm{F}}_{\mathrm{o}}(H)$ denote the focal subgroup of $H$ in $G$:

${\mathrm{F}}_{\mathrm{o}}(H)=⟨x{y}^{-1}\mid x,y\in H,\mathrm{\exists }g\in G,gx{g}^{-1}=y⟩$.

Then, we have:

$H\cap [G,G]={\mathrm{F}}_{\mathrm{o}}(H)$.