Isomorphisms of abelian Cayley graphs with their natural edge-colouring

We prove that if $\phi$ is an isomorphism between two connected Cayley graphs of abelian groups, and $\phi$ respects the natural edge-colourings of the Cayley graphs, then $\phi$ is the composition of a group isomorphism and a colour-preserving graph automorphism. This implies that if every colour-preserving automorphism of a connected abelian Cayley graph $Cay(G;S)$ is an affine map, then the same is true for every colour-permuting automorphism. We also show that this property holds if and only if the subgroup generated by $\{s\in S\mid 2s\mathrm{\ne }c\}\cup \{c\}$ has index $\le 2$ for every element $c$ of order $2$ in $G$.