Do you know this Burnside ring module? Let $G$ be a finite group and $\Omega(G)$ its Burnside ring. There is a certain $\Omega(G)$-module, let's call it $M(G)$, that appears in something that I am thinking about. As an abelian group $M(G)$ is the direct sum $\oplus_HR(W_GH)$, over conjugacy classes of subgroups $H\subset G$, of the Grothendieck group of complex representations of the Weyl group $W_GH=N_GH/H$. I don't have a lot of experience with $G$-spaces, but I would guess that this module has appeared before. My question is, does anybody know its name, or better yet does anybody know an illuminating way of thinking about it?
The module structure is as follows: let $K$ be a subgroup of $G$. Multiplication by the element of $\Omega(G)$ corresponding to the orbit $G/K$ is the composition of two maps $M(G)\to M(K)\to M(G)$. 
The map $M(G)\to M(K)$ takes the element $V\in R(W_GH)$ to the sum, over conjugacy classes of subgroups $L\subset K$ such that $L$ is conjugate to $H$ in $G$, of the restriction of $V$ along an injection $W_KL\to W_GH$. 
The map $M(K)\to M(G)$ takes the element $V\in R(W_KL)$ to the sum, over $N_KL$-conjugacy classes of subgroups $H\subset G$ such that $H\cap K=L$, of a representation of $W_GH$ obtained from $V$ by restricting and then inducing along injections $W_KL\leftarrow (N_GH\cap N_KL)/L\to W_GH$.
ADDED LATER: This turns out to be one of those cases where asking a question stimulates one to answer it oneself. But I would still be interested in references or further information.
 A: I've got it. Here's another way to think about $M(G)$. Let $\mathcal G(G)$ be the groupoid whose objects are the finite $G$-sets and whose morphisms are the isomorphisms. Let $Rep(G)$ be the additive category of all functors from $\mathcal G(G)$ to finite-dimensional complex vector spaces that are monoidal in the sense of taking disjoint union to direct sum. Then $M(G)$ is the Grothendieck group of $Rep(G)$. 
For a homomorphism $i:K\to G$ we get a forgetful functor $\mathcal G(G)\to \mathcal G(K)$ and its left adjoint, both of them monoidal. These induce additive functors from $Rep(K)\to Rep(G)$ and vice versa, hence homomorphisms $i_\ast: M(K)\to M(G)$ and $i^\ast: M(G)\to M(K)$, making $M$ both a covariant and a contravariant functor. 
For a finite $G$-set $X$, the monoidal functor $S\mapsto S\times X$ from $\mathcal G(G)$ to itself gives a map $M(G)\to M(G)$, and this is how $M(G)$ becomes a Burnside module. In the case when $X$ is $G/K$, this is the same as $i_\ast\circ i^\ast$ for the inclusion $i:K\to G$.
