Irreducible constituents of faithful induced characters

Let $G$ be a finite group, and suppose $\chi$is a character of $G$ obtained by inducing an irreducible character of some subgroup of $G$. If $\chi$ is faithful, we show that some irreducible constituent of $\chi$ has a solvable kernel. This yields an improved version of a theorem of Evdokimov and Ponomarenko.