I am wondering whether there is an easy example of a finite group $G$ with a Sylow $p$-subgroup $P$ and a subgroup $Q\leq P$ such that the normalizer $N_P(Q)$ of $Q$ in $P$ is NOT a Sylow $p$-subgroup of $N_G(Q)$, the normalizer of $Q$ in $G$.
Obviously when trying to find an example, you should look at non-commutative groups. Hence semi-direct products look like a good place to start. So I tried a couple of permutation groups and dihedral groups but failed to find an example. If I remember correctly I found out that such a group should have order at least 24.
To me this problem is not so important, but I would like to see an example. (Which according to some text I came across should be obvious).
Thanks in advance!