  We study the automorphisms of a Cayley graph that preserve its natural
edge-colouring. More precisely, we are interested in groups G, such that every
such automorphism of every connected Cayley graph on G has a very simple form:
the composition of a left-translation and a group automorphism. We find classes
of groups that have the property, and we determine the orders of all groups
that do not have the property. We also have analogous results for automorphisms
that permute the colours, rather than preserving them.
