Polynomial functors from free groups to a stable infinity-category

We study the category of polynomial functors from finitely generated free groups to a stable infinity-category D. We show that this category is equivalent to the category of excisive functors from pointed animas to D, and also to truncated right comodules over the commutative operad with values in D. The latter formulation generalizes a result of Geoffrey Powell in characteristic zero.
We use the equivalence of categories to calculate Ext between polynomial functors from free groups to abelian groups, extending previous results of Christine Vespa and others. Using the work of Aurelien Djament, we give applications to stable cohomology of automorphism groups of free groups with coefficients in a polynomial functor.