  We study higher Hochschild homology evaluated on wedges of circles, viewed as
a functor on the category of free groups. The principal results use
coefficients arising from square-zero extensions; this is motivated in part by
work of Turchin and Willwacher in relation to hairy graph cohomology.
  Working over a field of characteristic zero, achieve effective calculations
of higher Hochschild homology, generalizing initial results of Turchin and
Willwacher. This exploits exponential functors, which arise both in calculating
higher Hochschild homology and in relation to polynomial functors on free
groups. It is a crucial observation that these are related.
  For the main cases of interest, the action of automorphisms of free groups on
Hochschild homology factors across outer automorphisms. As the appropriate
underlying framework, we introduce and make essential use of the category of
outer functors, the full subcategory of functors on free groups on which inner
automorphisms act trivially.
