Generalized Burnside Algebra of type $B_n$

We define the generalized Burnside algebra $HB(W_n)$ for $B_n$-type Coxeter group $W_n$ and construct an surjective algebra morphism between Mantaci-Reutenauer algebra ${\sum }^{\mathrm{\prime }}(W_n)$ and $HB(W_n)$. Then, by obtaining the primitive idempotents $(e_{\lambda }{)}_{\lambda \in \mathcal{D}\mathcal{P}(n)}$ of $HB(W_n)$, we consider the image ${\text{res}}_{W_A}^{W_n}{e}_B$ and ${\text{ind}}_{W_A}^{W_n}{e}_B^A$ under restriction and induction map between generalized Burnside algebras. We give an alternative formula to compute the elements number of conjugate classes of $W_n$. We also obtain an effective method to determine the size of $\mathcal{C}(S_n)$ which is the set of elements of type $S_n$.