Quasi-monomial characters and the Fitting subgroup

An irreducible character of a finite group [Formula: see text] has been called quasi-monomial if it has a multiple that is monomial (induced from a linear character of some subgroup of [Formula: see text]). We show in this note that if [Formula: see text] is solvable, then a quasi-monomial character of [Formula: see text] is monomial if it restricts irreducibly to the Fitting subgroup of [Formula: see text] We give some related results for normal and subnormal subgroups of [Formula: see text]-groups (groups all of whose irreducible characters are monomial) and some examples of solvable groups having quasi-monomial characters that are not monomial.