Quadratic fields of values of finite solvable groups

We show that if $G$ is a finite solvable group such that $E=\mathbb{Q}(\{\chi (g)\mathrm{\mid }\chi \in \mathrm{I}\mathrm{r}\mathrm{r}(G),g\in G\})$ is an extension of degree 2 over the rationals, then $E=\mathbb{Q}(\sqrt{\pm d})$ for a positive integer $d$ all of whose prime divisors lie in $\{2,3,5\}$. Moreover, if $G$ has a composition factor of order 7, then either $E=\mathbb{Q}(\sqrt{2})$ or $E=\mathbb{Q}(\sqrt{-3})$. This provides a partial answer to a question of G. Navarro.