  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 automorphisms of the free group. These
representations show up in seemingly unrelated parts of mathematics, from
cohomology of moduli spaces of curves to polynomial functors on free groups and
Hochschild-Pirashvili cohomology.
  We show that these cohomology representations form a polynomial functor, and
use various geometric models to compute many of its composition factors. We
further compute the composition factors completely for all configurations of
$n\leq 10$ particles. An application of this analysis is a new
super-exponential lower bound on the symmetric group action on the weight $0$
component of $H^*_c(M_{2,n})$.
