Hall subgroup property I need help to prove a property regarding of Hall subgroups.
Let $N$ be a normal subgroup of $G$. If $T/N \in Hall_\pi(G/N)$ and $S \in Hall_\pi(T)$ then $S\in Hall_\pi(G)$
I have managed to show that for any Hall subgroup $H$ of $G$, $H\cap N$ is a hall subgroup of $N$ and $HN/N$ is a hall subgroup of $G/N$

Hint: $|G/N:T/N|=|G:T|$ and $|G:S|=|G:T|\cdot |T:S|$.
