Give an example of a group $G$ of order $p^2q$ where $p$ and $q$ are distinct prime numbers satisfying $q\not\equiv 1 \pmod{p}$ which does not have a unique $q$-Sylow subgroup.
My attempt:
Consider the dihedral group of 9-gon $D_9$. Then $O(D_9)=18=3^2×2$. Clearly $2 \not\equiv 1 \pmod{3}$. There are 9 axes of symmetry for 9-gon. So reflection about each axis gives a 2-Sylow subgroup. So we get at least 9 2-Sylow subgroups.
Will tHis example work?