  Given a rank 2 hermitian bundle over a 3-manifold that is non-trivial
admissible in the sense of Floer, one defines its Casson invariant as half the
signed count of its projectively flat connections, suitably perturbed. We show
that the 2-divisibility of this integer invariant is controlled in part by a
formula involving the mod 2 cohomology ring of the 3-manifold. This formula
counts flat connections on the induced adjoint bundle with Klein-four holonomy.
