Corollary 4

Let $N$ be a normal subgroup of $G$. The image in $G/N$ of a Sylow $p$-subgroup of $G$ is a Sylow $p$-subgroup of $G/N$ and every Sylow $p$-subgroup of $G/N$ is obtained in this way.

Let $G' = G/N$ and $P'$ be the image in $G'$ of a Sylow $p$-subgroup $P$ of $G$. The group $G$ operates transitively on $G'/P'$ and hence $G'/P'$ is equipotent to $G/S$, where $S$ is a subgroup of $G$ containing $P$. Therefore $(G':P')$ divides $(G:P)$, is thus not a multiple of $p$ and the $p$-group $P'$ is a Sylow $p$-subgroup of $G'$. Let $Q'$ be another Sylow $p$-subgroup of $G'$; then $Q' = g'P'{g'}^{-1}$ for some $g' \in G'$; if $g \in G$ is a representative of $g'$, the group $Q'$ is the image of $Q = gPg^{-1}$.