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We study the compactly supported rational cohomology of configuration spacesof points on wedges of spheres, equipped with natural actions of the symmetricgroup and the group $Out(F_g)$ of outer automorphisms of the free group. Theserepresentations show up in seemingly unrelated parts of mathematics, fromcohomology of moduli spaces of curves to polynomial functors on free groups andHochschild-Pirashvili cohomology.  We show that these cohomology representations form a polynomial functor, anduse various geometric models to compute many of its composition factors. Wefurther compute the composition factors completely for all configurations of$n\leq 10$ particles. An application of this analysis is a newsuper-exponential lower bound on the symmetric group action on the weight $0$component of $H^*_c(M_{2,n})$.

Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology