Colour-permuting automorphisms of complete Cayley graphs

Let $G$ be a (finite or infinite) group, and let $K_G=\mathrm{C}\mathrm{a}\mathrm{y}(G;G\setminus \{1\})$ 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 $\{s,s^{-1}\}$ for $s\in G$. We prove that every colour-permuting automorphism of $K_G$ is an affine map, unless $G\cong Q_8\times B$, where $Q_8$ is the quaternion group of order $8$, and $B$ is an abelian group, such that $b^2$ is trivial for all $b\in B$. We also prove (without any restriction on $G$) that every colour-permuting automorphism of $K_G$ 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.