Inducing irreducible $B_n \times S_k$ characters to $B_{n+k}$ I know that we can induce irreducible representations of $B_n$ to $B_{n+k}$ using the Ariki-Koike branching rule.
The irreducible representations of $B_n \times S_k$ are parametrised by tuples 2-partitions of $n$ and partitions of $k$.
I just haven't found an analogue to the branching rule for the these types of subgroups.
 A: Expanding on my comment, here's what the two steps look like combinatorially.
Induction from $S_k$ to $B_k$ sends $S^\lambda$ to $\bigoplus_{\mu, \gamma} W(\mu,\gamma)^{\oplus c^\lambda_{\mu,\gamma}}$ where $c^\lambda_{\mu,\gamma} $ denotes a Littlewood–Richardson coefficient.  This I think is easy to see using Frobenius reciprocity and the fact that $W(\mu,\gamma)$ restricts to $\bigoplus_\lambda (S^\lambda)^{c^\lambda_{\mu,\gamma}}$, which is pretty clear from the construction of the $W(\mu,\gamma)$'s.
Induction from $B_n \times B_k$ to $B_{n+k}$ is also governed by Littlewood–Richardson coefficients essentially independently in the two factors  —  this is true for all wreath products $G \wr S_n$.  Inducing $W(\mu,\gamma) \otimes W(\alpha,\beta)$ from $B_n \times B_k$ to $B_{n+k}$ decomposes as $\bigoplus W(\lambda, \tau) ^{c^\lambda_{\mu,\alpha}c^\tau_{\gamma,\beta}}$.  This can probably be found anywhere that talks about representations of wreath products, I am particularly fond of Zelevinski's book on positive self adjoint Hopf algebras for this stuff.
