Colour-permuting automorphisms of complete Cayley graphs

Let G be a (finite or infinite) group, and let KG = Cay ( G;G \1\ ) be the complete graph with vertex set G, considered as a Cayley graph of G. Being a Cayley graph, it has a natural edge-colouring by sets of the form , s⁻¹\ for s ∈ G. We prove that every colour-permuting automorphism of KG is an affine map, unless G Q₈ × B, where Q₈ is the quaternion group of order $8$, and B is an abelian group, such that b² is trivial for all b ∈ B. We also prove (without any restriction on G) that every colour-permuting automorphism of KG is the composition of a group automorphism and a colour-preserving graph automorphism. This was conjectured by D.P.Byrne, M.J.Donner, and T.Q.Sibley in 2013.