Let $G$ be a finite group and let $N$ be a normal subgroup of $G$ with the property that $G/N$ is nilpotent. Prove that there exists a nilpotent subgroup $H$ of $G$ satisfying $G = HN$.
This is problem 223 on page 24 of http://www.math.kent.edu/~white/qual/list/all.pdf. I think that Frattini's argument may be useful.
Use induction on $|G|$. If all Sylow $p$-subgroups of $G$ are normal, then $G$ is nilpotent. Otherwise, there exists a non-normal Sylow $p$-subgroup $P$. Apply the Frattini argument to $PN \unlhd G$ to get $G=NN_G(P)$ and then apply induction to $N_G(P)$.