Twisted actions and the obstruction to extending unitary representations of subgroups

Suppose that $G$ is a locally compact group and $\pi$ is a (not necessarily irreducible) unitary representation of a closed normal subgroup $N$ of $G$ on a Hilbert space $H$. We extend results of Clifford and Mackey to determine when $\pi$ extends to a unitary representation of $G$ on the same space $H$ in terms of a cohomological obstruction.