On a question of Navarro on the field of values of characters of solvable groups

In a 2023 survey paper, Gabriel Navarro posed the following problem: Given an irreducible character $\chi$ of some solvable group $G$ where $\chi$ either is 2-rational or has odd degree, does there exist some $g\in G$ such that $\mathbb{Q}(\chi (g))=\mathbb{Q}(\chi )$? When $\chi$ is imprimitive, the answer is no in general. In the case where $\chi$ is primitive, we are able to show that there exists $g\in G$ such that $Gal(\mathbb{Q}(\chi )\mathrm{/}\mathbb{Q}(\chi (g)))$ is an elementary abelian 2-group. We are then able to show that the answer to Navarro’s problem is yes when we further assume that $c(\chi )$ is divisible by at most two primes.