Modular characters, hall subgroups, and normal complements

If H is a Hall subgroup of a finite group G, it was proven in 1989 using the classification of finite simple groups that all the irreducible complex characters of H extend to G if and only if there is \(N\trianglelefteq \,G\) $N⊴G$ such that \(HN=G\) $HN=G$ and \(H\cap N=1\) $H\cap N=1$ . In this note we offer a modular version of this result under more general hypotheses.