Automorphic subsets of the n-dimensional cube

An automorphic subset of the n-dimensional cube Qn is an orbit of a subgroup of Aut(Qn), acting on the vertices. We develop a theory of such subsets, and we show that those containing 0 coincide with the cwatsets introduced by Sherman and Wattenberg in response to a statistical result of Hartigan. Using this characterisation, together with results from finite group theory and number theory, we answer two questions on cwatsets posed by Sherman and Wattenberg, and we complete the proofs of some results outlined by Kerr.