Finite linear groups containing an irreducible solvable normal subgroup

The following theorem is proved. Let $G$ be a finite group which has a faithful representation $X$ of degree $n$ over the complex number field such that $X|H$ is irreducible where $H$ is a solvable normal subgroup of $G$. Let $p$ be a prime and assume that $n$ is neither a multiple of $p$ nor a multiple of a prime power ${q^s}$ with ${q^s} \equiv \pm 1\;\bmod \;p$. Then a $p$-Sylow subgroup of $G$ is normal and abelian.