In 'Finite Groups' by Gorenstein, it is stated that if $P$ is a Sylow $p$-subgroup of $G$ then there exists a normal subgroup $K$ such that $G/K$ is isomorphic to $P/P \cap G'$′. The proof is the following:
$P \cap G'$ is a Sylow $p$-subgroup of $G'$, as $G/G'$ is abelian and $G' \triangleleft G$. If $K$ denotes the inverse image of $O_{P'}(G/G')$ in $G$, $P \cap G' = P \cap K$, $K \triangleleft G$ and $G/K$ is an abelian $p$-group isomorphic to $P/P \cap G'$.
I understand the first line, and that $O_{P'}(G/G')$ represents the maximal normal $p'$-subgroup of $G/G'$ but don't quite understand what the inverse image is, or how this immediately yields $P \cap G' = P \cap K$, could anyone explain this in better terms?