For each integer $k\geq 3$, give an example of a finite group $G$ and a subgroup $H$ such that $|G|=k|H|$ and $H$ is not normal in $G$.
I think the case that $k=|G|$ is not considered here.
I have doubts about the way this question is asked; by Lagrange's theorem, not every $3\leq k<|G|$ can give a respective subgroup.