Hello /sci/,
I have written a new post at Psychic Apparatuses (https://psychicapparatuses.wordpress.com/2017/06/11/action-complex-of-homotopy-groups/) detailing the construction of a complex of commuting actions of lower homotopy groups on higher homotopy groups. I haven't found anything in the literature discussing these higher actions, but it's all quite natural since all of the actions satisfy nice universal properties with respect to one another (the group structure on connected components of iterated loop spaces is just a special subsystem of this complex).
Any thoughts on what can be done with this? How do commuting actions restrict the structure of the groups involved?