In this paper, we continue our study of the pro-Σ fundamental groups of configuration spaces associated to a hyperbolic curve, where Σ is either the set of all prime numbers or a set consisting of a single prime number, begun in an earlier paper. Our main result may be regarded either as a combinatorial, partially bijective generalization of an injectivity theorem due to Matsumoto or as a generalization to arbitrary hyperbolic curves of injectivity and bijectivity results for genus zero curves due to Nakamura and Harbater-Schneps. More precisely, we show that if one restricts one’s attention to outer automorphisms of such a pro-Σ fundamental group of the configuration space associated to a(n) affine (respectively, proper) hyperbolic curve which are compatible with certain “fiber subgroups ” [i.e., groups that arise as kernels of the various natural projections of a configuration space to lower-dimensional configuration spaces] as well as with certain cuspidal inertia subgroups, then, as one lowers the dimension of the configuration space under consideration from n +1 to n ≥ 1 (respectively, n ≥ 2), there is a natural injection between the resulting groups of such outer automorphisms, which is a bijection if n ≥ 4. The key tool in the proo
