I'm interested in computing sums like $\sum_{\sigma \in W} tr(\sigma ^3)$ , where $W$ is the Weyl group of $SO(2n+1)$, i.e. $W = (\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$. I tried to figure out what an element of $W$ looks like represented as a matrix but got in trouble when trying to compute the cubes and traces. I guess it will turn out to be quite a complicated sum over all partitions of $n$ and n-tuples of elements in $\mathbb{Z}/2$, but I'm stuck.
What is a good way to go about this problem? Does somebody see if there is an elegant solution?