We study the compactly supported rational cohomology of configuration spaces of points on wedges of spheres, equipped with natural actions of the symmetric group and the group Out(F g ) of outer automorphism of the free group. These representations are closely related to Hochschild-Pirashvili homology with coefficients in square-zero algebras, and they show up in seemingly unrelated parts of mathematics, from cohomology of moduli spaces of curves to polynomial functors on free groups.We show that these cohomology representations form a polynomial functor, and use various geometric models to compute a substantial part of its composition factors. We further compute the composition factors completely for all configurations of n ≤ 10 particles. An application of this analysis is a new super-exponential lower bound on the symmetric group action on the weight 0 component of H * c (M 2,n ).