Essential algebra of the shifted Burnside biset functor with abelian shift
Let $T$ be a finite group and let $kB_T$ denote the shifted Burnside biset functor over a commutative ring $k$. We study the essential algebra $\widehat{kB_T}(G)$ of $kB_T$ at an arbitrary finite group $G$. We first establish criteria determining when a covering subgroup of $G\times G\times T$ factors, with respect to the star product, through a group of order strictly smaller than $|G|$; the first criterion places no hypothesis on $T$. As an application, we describe $\widehat{kB_T}(G)$ completely whenever no nontrivial quotient of $G$ is isomorphic to a section of $T$, extending the coprime case treated by Romero. When $T$ is abelian and $|T|$ is invertible in $k$, we prove that $\widehat{kB_T}(G)$ decomposes as a direct sum of matrix algebras over the group algebras of the groups $Out_T(A)$, where $A$ runs over the linkage classes of reduced subgroups of $G\times T$, and we parametrize the simple modules of $\widehat{kB_T}(G)$.