On quasi-primitive characters of solvable groups

Let chi is an element of Irr(G) be a quasi-primitive character with odd degree, and suppose that G is pi(chi(1))-solvable group. Wilde associated to chi a unique conjugacy class of subgroups U subset of G satisfying chi(chi) over bar = (1(U))(G). We construct in this situation a sequence of character pairs (U-i, xi(i)), where xi(i) is an element of Irr(U-i) is quasi-primitive and each (U-i, xi(i)) is uniquely determined up to conjugacy in G, such that G = U-0 > U-1 > . . . > U-n, and chi(1) = xi(0)(1) > xi(1)(1) > . . . > xi(n)(1) = 1. Furthermore, we have chi(chi) over bar = (xi(i)(xi) over bar (i))(G) for each i, and in particular chi(chi) over bar = (1(Un))(G). We also prove that the subgroups U-n and U are conjugate in G, and thus present a new description for Wilde's result.