Let G be a group of order $|G|=pq^m$ where $p, q$ are primes with $q^m<p$.
Use a Sylow counting argument to show that $G\cong C_p \rtimes_h Q$ where $Q$ is a group with $|Q|=q^m$ and $h:Q\rightarrow Aut(C_p)$ is a homomorphism
I've shown that $C_p$ is a normal subgroup in $G$, but I'm not sure how to proceed from there.