Solvable groups whose monomial, monolithic characters have prime power codegrees

In this note, we prove that if $G$ is solvable and $\mathrm{c}\mathrm{o}\mathrm{d}(\chi )$ is a $p$-power for every nonlinear, monomial, monolithic $\chi \in \mathrm{I}\mathrm{r}\mathrm{r}(G)$ or every nonlinear, monomial, monolithic $\chi \in \mathrm{I}\mathrm{B}\mathrm{r}(G)$, then $P$ is normal in $G$, where $p$ is a prime and $P$ is a Sylow $p$-subgroup of $G$.