If $p < q$ are primes then there is a nonabelian group of order $pq$ iff $q = 1 \pmod p$, in which case the group is unique. If $p = 2$ we obtain the dihedral group of order $2q$, which generalizes first to the dihedral group of order $2n$ and then even further to the "generalized dihedral group" where the cyclic group of order $n$ is replaced with any abelian group.

What if $p > 2$? Is there a natural generalization of the groups of order $pq$ to a family of groups of order $pn$? Maybe more than one possible generalization? Is it maybe even meaningful to talk about the "generalized $p$-hedral group"?