Suppose we have a finite group $A$ which is the semidirect product of the normal subgroup $G$ by a complement $H$ or another complement $K$: $A = G \rtimes H = G \rtimes K$.
Now fix a prime $p$ dividing the order of $G$ and the one of $H$, and consider $P_A, P_G, P_H$ and $P_K$ the (non-trivial) $p$-Sylow subgroups of the respective groups, and suppose that $P_K \leq P_A = P_G \times P_H$. As $H \cong K$, we also have $P_H \cong P_K$. I also know that $|P_H| \leq |P_G|$.
First, what can we say about the relationship between $P_H$ and $P_K$? Can they intersect trivially? Are they conjugate? Are any of their subgroups conjugate?
Secondly, is it possible in this setting that $P_K$ doesn't normalize any $p$-Sylow of $A$?
Thirdly: can we find a subgroup $P_G' < P_G$ such that $P_K \leq P_G' \times P_H$?
An answer to any one of these three questions would be great.
I don't know if it helps but we can also suppose that the whole of $H$ centralizes $P_G$, and eventually that all the Sylows are abelian, but I am treating two separate cases and the second hypothesis only works in one of the two, so I'd rather not use that.