Action of automorphisms of pure braid groups on homotopy groups of two-sphere

We examine the Moore complex of the Δ-group structure related to the pure braid groups and introduced by Berrick, Cohen, Wong, and Wu. We prove that the cycle and the boundary groups are invariant under all automorphisms of the pure braid groups, and thereby, we extend the results of Li and Wu on the reflection automorphism. We conclude that there is an induced action of all automorphisms of the pure braid groups on the homotopy groups of the two-sphere. Besides, we compute this action for a small number of strands.