On prime divisors of character degrees and codegrees

Let $G$ be a finite group, and let $\mathrm{I}\mathrm{r}\mathrm{r}(G)$ denote the set of irreducible complex characters of $G$. For $\epsilon \in \{\pm \}$, we define ${\mathrm{c}\mathrm{d}}_{\epsilon }(G)=\{{\chi }_{\epsilon }(1)\mid \chi \in \mathrm{I}\mathrm{r}\mathrm{r}(G)\}$, where ${\chi }_{+}(1)=\chi (1)$ denotes the degree of $\chi$, ${\chi }_{-}(1)=\mathrm{\mid }G:\ker (\chi )\mathrm{\mid }\mathrm{/}\chi (1)$ denotes the codegree of $\chi$. Further, let ${\omega }_{\epsilon }(G)=\{\pi (n)\mid n\in {\mathrm{c}\mathrm{d}}_{\epsilon }(G)\}$, where $\pi (n)$ stands for the set of prime divisors of $n$. We established that if $\mathrm{\mid }{\omega }_{\epsilon }(G)\mathrm{\mid }\le 3$, then $G$ is solvable. Additionally, a generalization of this result is obtained in the case when $\epsilon =+$.