The functor of p-permutation modules for abelian groups

Let k be a field of characteristic p, where p is a prime number, let pp_k(G) be the Grothendieck group of p-permutation kG-modules, where G is a finite group, and let Cpp_k(G) be pp_k(G) tensored with the field of complex numbers C. In this article, we find all the composition factors of the biset functor Cpp_k restricted to the category of abelian groups.