  We give a Type $B$ analog of Whitehouse's lifts of the Eulerian
representations from $S_n$ to $S_{n+1}$ by introducing a family of
$B_{n}$-representations that lift to $B_{n+1}$. As in Type $A$, we interpret
these representations combinatorially via a family of orthogonal idempotents in
the Mantaci-Reutenauer algebra, and topologically as the graded pieces of the
cohomology of a certain $\mathbb{Z}_{2}$-orbit configuration space of
$\mathbb{R}^{3}$. We show that the lifted $B_{n+1}$-representations also have a
configuration space interpretation, and further parallel the Type $A$ story by
giving analogs of many of its notable properties, such as connections to
equivariant cohomology and the Varchenko-Gelfand ring.
