  Let $N_k$ denote the closed non-orientable surface of genus $k$. In this
paper we study the behaviour of the `square map' from the group of isometries
of hyperbolic 3-space to the subgroup of orientation preserving isometries. We
show that there are $2^{k+1}$ connected components of representations of
$\pi_1(N_k)$ in Isom$(\mathbb{H}^3)$, which are distinguished by the
Stiefel-Whitney classes of the associated flat bundle.
