I need help with the following exercise, which, judging by its position in the book, should follow more or less directly from the Sylow theorems.
Let G,G' be two finite groups and $\phi$ a homomorphism of $G$ onto G'. Let $p$ be a prime number and $P,P_1$ two Sylow $p$-subgroups of $G$ such that $\phi(P)=\phi(P_1)$. Show that there exists $x\in\text{Ker}(\phi)$ such that $P=xP_1x^{-1}$.
Well, I know that there exists $y\in G$ such that $P=yP_1y^{-1}$ and it's easy to see that the set of $x$ such that $P=xP_1x^{-1}$ is $yN_G(P_1)$. So we have to show $yN_G(P_1)\cap\text{Ker}(\phi)\neq\emptyset.$
But what now? A hint would be most welcome.