Sylow subgroups are in correspondence with Sylow subgroups of quotient by central subgroup

Statement
Suppose $$G$$ is a finite group and $$H$$ is a fact about::central subgroup of $$G$$, i.e., $$H$$ is contained in the fact about::center of $$G$$. Let:

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

be the projection map. Then, there is a natural bijection between the fact about::Sylow subgroups of $$G$$ and the Sylow subgroups of $$G/H$$, given by:


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

In the special case that the central subgroup is the full center of $$G$$, the Sylow subgroups of $$G$$ are in bijection with the Sylow subgroups of the fact about::inner automorphism group of $$G$$.

Related facts

 * Sylow subgroups are in correspondence with Sylow subgroups of quotient by hypercenter
 * Equivalence of definitions of finite nilpotent group
 * Every Sylow subgroup intersects the center nontrivially or is contained in a centralizer