Kernels of minimal characters of solvable groups

Let $G$ be a finite solvable group. We prove that if $\chi \in \mathrm{I}\mathrm{r}\mathrm{r}(G)$ has odd degree and $\chi (1)$ is the minimal degree of the non-linear irreducible characters of $G$, then $G\mathrm{/}\mathrm{K}\mathrm{e}\mathrm{r}\chi$ is nilpotent-by-abelian.