We present a homotopy-theoretic method for calculating Ext groups between polynomial functors from the category of (finitely generated, free) groups to abelian groups. It enables us to substantially extend the range of what can be calculated. In particular, we can calculate torsion in the Ext groups, about which very little has been known. We will discuss some applications to the stable cohomology of Aut(F_n), based on a theorem of Djament.