Embed the hyperoctahedral group $H_n$ into the symmetric group $S_{2n}$ as the centralizer of the involution $(1, 2) (3, 4) \cdots (2n-1, 2n)$ (cycle notation). Label representations of $S_{2n}$ by partitions of $2n$ and label representations of $H_n$ by pairs of partitions whose sizes add up to $n$ in the standard way. I am looking for a combinatorial description of the branching rule from $S_{2n}$ to $H_n$. This should be in the literature, but I couldn't find it.

Second small question: Is it possible to have a different embedding of $H_n$ into $S_{2n}$ so that the rule changes? I'd guess probably, but didn't think hard about it.