On Automorphisms of Irreducible Linear Groups with an Abelian Sylow $2$-Subgroup

Let $\Gamma=AG$ be a finite group, where $G\triangleleft\Gamma$, $(|G|,|A|)=1$, and let $A$ be a nonprimary subgroup of odd order which is not normal in $\Gamma$. The Sylow $2$-subgroup of the group $G$ is Abelian, and $C_G(a)=C_G(A)$ for every element $a\in A^{\#}$, where $A^{\#}$ stands for the set of nonidentity elements of $A$. Suppose that the group $G$ has a faithful irreducible complex character of degree $n$ which is $a$-invariant for at least one element $a\in A^{\#}$. In the present paper, it is proved that $n$ is divisible by a power of a prime with exponent $f>1$ such that $f\equiv -1$ or $1\,(\operatorname{mod}|A|)$.