Consider finite groups of the form $G=\langle g,s \rangle$ generated by two elements, such that $s$ is order $p$, and it generates the (cyclic) sylow $p$-subgroup of $G$, $g$ is of order coprime to $p$, and there exists a normal subgroup $H$ of $G$ such that $H$ is index $p$, so we have a normal $p$ complement.
Is this family of groups "classifiable"? I am being purposely loose with the word classifiable here, even if a precise classification isn't possible, can we say anything precise about the internal structure of $G$?
Any references that deal with the structure of such groups would also be much appreciated, I only know theorems that give existence of $p$ complements, but they don't really address the internal structure.