The group of endo-permutation modules

The group D(P) of all endo-permutation modules for a finite p-group P is a finitely generated abelian group. We prove that its torsion-free rank is equal to the number of conjugacy classes of non-cyclic subgroups of P. We also obtain partial results on its torsion subgroup. We determine next the structure of Q\otimes D(-) viewed as a functor, which turns out to be a simple functor S_{E,Q}, indexed by the elementary group E of order p^2 and the trivial Out(E)-module Q. Finally we describe a rather strange exact sequence relating Q\otimes D(P), Q\otimes B(P), and Q\otimes R(P), where B(P) is the Burnside ring and R(P) is the Grothendieck ring of QP-modules.