The Maximal Number of Iπ-Characters Over Normal Subgroups

Let G be a π-separable group for a set of primes, let N be a normal subgroup of G, and let θ be an Iπ-character (i.e., irreducible π-partial character) of N. We obtain a necessary and sufficient condition for the number of Iπ-characters of G over θ to take the possible maximum |G:N|π. Some applications are given.