On quadratic rational Frobenius groups

Let G be a finite group and, for a given complex character (Formula presented.) of G, let (Formula presented.) denote the field extension of (Formula presented.) obtained by adjoining all the values (Formula presented.), for (Formula presented.). The group G is called quadratic rational if, for every irreducible complex character (Formula presented.), the field (Formula presented.) is an extension of (Formula presented.) of degree at most 2. Quadratic rational groups generalize the well-known concept of a cut group. In this paper we classify the Frobenius groups that are quadratic rational.