The Dade group of Mackey functors for p-groups

We introduce endo-permutation Mackey functors and the Mackey-Dade group of a $p$-group and study their basic properties. We exhibit relations between the Mackey-Dade group of a finite $p$-group $P$ and the Dade groups of the Weyl groups of all subgroups of $P$. As a result, we show that the Mackey-Dade group tensored with $\mathbb{Q}$ over $\mathbb{Z}$ is the kernel of the linearization map for Mackey functors, introduced by the author.