  We give an explicit decomposition of $\hbox{Ind}(1)_{B_n}^{S_{2n}}$,
following Barbasch and Vogan [1]. We define two natural generalizations of
$B_n$, and extend the proof in [1] to recursively compute these decompositions.
Although the decompositions do not appear to follow a simple pattern, we prove
enough of their structure to show that they are almost never multiplicity-free.
