On fully ramified Brauer characters

Let Z   be a normal subgroup of a finite group, let p≠5p≠5 be a prime and let λ∈IBr(Z)λ∈IBr(Z) be an irreducible G-invariant p-Brauer character of Z  . Suppose that λG=eφλG=eφ for some φ∈IBr(G)φ∈IBr(G). Then G/ZG/Z is solvable. In other words, a twisted group algebra over an algebraically closed field of characteristic not 5 with a unique class of simple modules comes from a solvable group.