Let $p$ be a prime number, $S$ be a cyclic group such that $p\nmid |S|$, $P$ a $p$-group, and let $S\to H\to P$ be a group extension diagram.
I have read in this paper by Segal that such an extension is always split if the groups involved are finite, but since I am a novice in finite group theory I was unable to come up with a proof yet.
Further more, Assume we had morphisms from $H$ and $P$ to $C_2$ such that the projection commutes with them, and the image of $S$ in $H$ is sent to $0$. Can one then pick a section that commutes with these morphisms as well?
Edit: I just realized that the second part of this question is true, and so pretty easily. Let $\phi:H\to C_2$, $\psi:P\to C_2$ and call the projection $\alpha:H\to P$, and assume $\phi=\psi\circ\alpha$. Given a section $\beta:P\to H$, that is $\alpha\circ\beta=\mathrm{id}$, then $\phi\circ\beta=\psi\circ\alpha\circ\beta=\psi$.
Also, I wanted to clarify the case I am interested in. $S$ is a cyclic subgroup of $G$ and we consider the normalizer $N_S$ and the Weyl group $W_S=N_S/S$. For $P$ a Sylow $p$-subgroup of $W_S$ and $H$ its preimage in $N_S$, is the extension $S\to H\to P$ split?