Let $G$ be a group of order $2\cdot 17\cdot 23$. Let $P$ be a $17$-Sylow subgroup and $Q$ a $23$-Sylow subgroup. Show that $PQ$ is normal in $G$ and that $P$ is characteristic in $PQ$.
What I know: both $P$ and $Q$ are normal in $G$ by the Sylow theorems, $PQ$ is normal in $G$ because it has index 2.
So all that remains is to show that $P$ is characteristic in $PQ$. Now, certainly any inner automorphism of $PQ$ will fix $P$ since $17$-Sylow subgroups are all conjugate and $P$ is normal in $G$ (and hence unique). But how do I know there isn't some outer automorphism that pushes $P$ around?
More generally, it seems like this would be true for groups of order $2pq$ for primes $2<p<q$, since I doubt $17$ and $23$ are particularly significant. Thoughts?