Existence of a normal p-Sylow subgroup when $\lvert\mkern1mu G\mkern1mu \rvert=q^{2}\mkern1mu p$

Question

If $\lvert\,G\,\rvert=p^{2}\cdot\,q$ where $q,p$ are primes, How can i show $G$ has a normal p-sylow subgroup ? I tried working with the sylow theorem but i cant reach any contradiction when $Np = q$ and $Nq = p,p^2$ Also tried searching the same question but couldn't find in Stack Exchange

Answer 1

This is false for $q=2, p=3$. The $3$-Sylow subgroups of $A_4$ are not normal: there are four copies of $A_3$ in $A_4$.