Sylow subgroups are in correspondence with Sylow subgroups of quotient by hypercenter

Statement
Suppose $$G$$ is a finite group and $$H$$ is the fact about::hypercenter of $$G$$. Let:

$$\pi:G \to G/H$$

be the natural quotient map.

Then, there is a natural correspondence between the fact about::Sylow subgroups of $$G$$ and the Sylow subgroups of $$G/H$$, given as follows:


 * A Sylow subgroup $$P$$ of $$G$$ corresponds to its image $$\pi(P)$$ in $$G/H$$.
 * A $$p$$-Sylow subgroup $$Q$$ of $$G$$ corresponds to the unique $$p$$-Sylow subgroup in $$\pi^{-1}(Q)$$.