Primitive characters of maximal subgroups of solvable groups

Let G be an M-group, and suppose that H is a maximal subgroup of odd index and that β∈IrrH is primitive. It was shown by Mark Lewis that β2(1) divides |G:H|. We show in this situation that either β2(1)=|G:H| or β(1)=1. A number of related results are established for groups that are not necessarily M-groups.