Character codegrees, kernels, and Fitting heights of solvable groups

For an irreducible character $\chi$ of a finite group $G$, let $\mathrm{cod}(\chi):=|G: \ker(\chi)|/\chi(1)$ denote the codegree of $\chi$, and let $\mathrm{cod}(G)$ be the set of irreducible character codegrees of $G$. In this note, we prove that if $\ker(\chi)$ is not nilpotent, then there exists an irreducible character $\xi$ of $G$ such that $\ker(\xi)<\ker(\chi)$ and $\mathrm{cod}(\xi)> \mathrm{cod}(\chi)$. This provides a character codegree analogue of a classical theorem of Broline and Garrison. As a consequence, we obtain that for a nonidentity solvable group $G$, its Fitting height $\ell_{\mathbf{F}}(G)$ does not exceed $|\mathrm{cod}(G)|-1$. Additionally, we provide two other upper bounds for the Fitting height of a solvable group $G$ as follows: $\ell_{\mathbf{F}}(G)\leq \frac{1}{2}(|\mathrm{cod}(G)|+2)$, and $\ell_{\mathbf{F}}(G)\leq 8\log_2(|\mathrm{cod}(G)|)+80$.