  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.
