Let $W_n$ be a Coxeter group of type $B_n$ with $n\geq 1$. Concretely, it is generated by a set of simple reflexions $S = \{s_1,\ldots ,s_n\}$ which satisfy the relations $s_i^2 = 1, s_is_j=s_js_i$ as long as $|i-j| \geq 2$, $(s_is_{i-1})^3 = 1$ for $2\leq i \leq n-1$ and $(s_ns_{n-1})^4 = 1$.

Given integers $a,b \geq 0$ such that $a+b = n$, consider the subgroup $H_{a,b} \simeq \mathfrak S_a \times W_b \subset W_n$ which is generated by all the simple reflexions except $s_a$.

Complex characters of the symmetric group $\mathfrak S_n$ (resp. of the group $W_n$) are classically classified by partitions of $n$ (resp. by bipartitions of $n$). Any such representation will be denoted by the letter $\rho$ with index a (bi)partition. Given a partition $\lambda \vdash a$ and a bipartition $(\alpha,\beta) \vdash b$, is there any paper or textbook reference describing the Littlewood-Richardson type formula to compute the induced character
$$\mathrm{Ind}_{H_{a,b}}^{W_n} \, \rho_{\lambda}\otimes \rho_{\alpha,\beta}?$$
In the special case where $\lambda = (a)$ is the trivial partition, this induction corresponds to Pieri's rule for groups of type $B_n$, which is described in paragraph 6.1.9 of Geck and Pfeiffer's book *Characters of finite Coxeter groups and Iwahori-Hecke algebras* (2000). In terms of Young diagrams, the rule says that we obtain the multiplicity-free sum of all the irreducible characters $\rho_{\alpha',\beta'}$ where, for some $0 \leq d \leq a$, $\alpha'$ (resp. $\beta'$) can be obtained from $\alpha$ (resp. $\beta$) by successively adding $d$ boxes (resp. $a-d$ boxes) in the Young diagram, with no two boxes in the same column.

If anything, I am particularly interested in the case where $\lambda$ is a hook partition, ie. $\lambda = (x,1^{a-x})$ for some $1\leq x \leq a$.