The hyperoctahedral group $H_n$ has several descriptions; as a wreath product; as signed permutation matrices; as the Weyl group of type $B_n$ or $C_n$. In all these descriptions it is apparent that the symmetric group $S_n$ is a subgroup.

I would like to know the Frobenius characters of the restrictions of the irreducible characters of $H_n$ to $S_n$. I imagine this is known but not to me.

The irreducible characters of $H_n$ are indexed by pairs of partitions $\alpha$,$\beta$ such that the total number of boxes is $n$. There is a well developed combinatorial theory involving bitableaux as well as an analogue of the Robinson-Schensted correspondence. The character theory is described in I. G. MacDonald "Symmetric functions and Hall polynomials" Chapter I, Appendix B in terms of symmetric functions.

I am also aware of the question 48532 which asks about the restriction map for the inclusion of $H_n$ in $S_{2n}$.