Cohomology of Burnside Rings

Let $G$ be a finite group and $A(G)$ its Burnside ring. For $H\subset G$ let ${\mathbb{Z}}_H$ denote the $A(G)$-module corresponding to the mark homomorphism associated to $H$. When the order of $G$ is square-free we give a complete description of the $A(G)$-modules $$\text{Ext}$^l_{A(G)}($\mathbb{Z}$_H, $\mathbb{Z}$_J)$and$$\text{Tor}$^{A(G)}_l($\mathbb{Z}$_H, $\mathbb{Z}$_J)$forany$H, J \subset G$and$l \geq 0$\mathrm{.}Weshowthatiftheorderof$G$ is not square-free then there exist $H,J\subset G$ such that ${\text{Ext}}_{A(G)}^l({\mathbb{Z}}_H,{\mathbb{Z}}_J)$ and ${\text{Tor}}_l^{A(G)}({\mathbb{Z}}_H,{\mathbb{Z}}_J)$ have unbounded rank as finite groups.