Character covering number of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(q)$

For a group $G$ and a character $\chi$ of $G$, let $c(\chi )$ denote the set of all irreducible characters of $G$, occurring in $\chi$. We prove that whenever $q\ge 8$, all non-trivial irreducible character $\chi$ of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(q)$ satisfies $c({\chi }^4)=\mathrm{I}\mathrm{r}\mathrm{r}({\mathrm{P}\mathrm{S}\mathrm{L}}_2(q))$ if $q=2^{2m+1}$ and $c({\chi }^3)=\mathrm{I}\mathrm{r}\mathrm{r}({\mathrm{P}\mathrm{S}\mathrm{L}}_2(q))$ otherwise.