Schur–Zassenhaus exercise: coprime subgroup contained in a complement
Let $K$ be a complement to $N$ in $G$, which exists by Schur-Zassenhaus. Then $|K|=|G|/|N|$. Consider the subgroup $UN$. Since $K(UN)=G$, $|K\cap UN|=|K|\cdot |UN|/|G|=|K|\cdot|U|\cdot|N|/|G|=|U|$, since $K$ is a complement to $N$. Thus, $K\cap UN$ is a complement to $N$ in $UN$. Since either $U$ or $N$ is solvable, all complements to $N$ in $UN$ are conjugate, so $K\cap UN=U^g$. Thus, $K^{g^{-1}}$ contains $U$.
Steve