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.