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A Cayley graph on a group $G$ has a natural edge-colouring. We say that sucha graph is CCA if every automorphism of the graph that preserves thisedge-colouring is an element of the normaliser of the regular representation of$G$. A group $G$ is then said to be CCA if every Cayley graph on $G$ is CCA.  Our main result is a characterisation of non-CCA graphs on groups that areSylow cyclic and whose order is not divisible by four. We also provide severalnew constructions of non-CCA graphs.

Characterising CCA Sylow cyclic groups whose order is not divisible by four