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 $N⊴G$ such that $HN=G$ and $H\cap N=1$. In this note we offer a modular version of this result under more general hypothesis.