On irreducible linear groups of prime-power degree

Let $\Gamma=AG$ be a finite group, $G\triangleleft\Gamma$, $(|A|,|G|)=1$, $C_G(a)=C_G(A)$ for each element $a\in A^{\#}$, and let the subgroup $A$ have a nonprimary odd order and be not normal in $\Gamma$. Assume that $\chi$ is an irreducible complex character of $G$ that is invariant for at least one nonunity element of $A$ and $\chi(1)<2|A|$. Then it is proved that $G=O_q(G)C_G(A)$ and $\chi(1)$ is a power of a prime $q$. Furthermore, if $G$ is not solvable, then $\chi(1)=2(|A|-1)$ and $C_G(A)/Z(\Gamma)\cong PSL(2,5)$.