Analogue of focal subgroup theorem for Hall subgroups

Statement
Suppose $$G$$ is a finite group and $$H$$ is a fact about::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:

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