A new canonical induction formula for $p$-permutation modules

Applying Robert Boltje's theory of canonical induction, we give a restriction-preserving formula expressing any $p$-permutation module as a $\mathbb{Z}[1/p]$-linear combination of modules induced and inflated from projective modules associated with subquotient groups. The underlying constructions include, for any given finite group, a ring with a $\mathbb{Z}$-basis indexed by conjugacy classes of triples $(U, K, E)$ where $U$ is a subgroup, $K$ is a $p'$-residue-free normal subgroup of $U$ and $E$ is an indecomposable projective module of the group algebra of $U/K$.