  A Cayley graph on a group $G$ has a natural edge-colouring. We say that such
a graph is CCA if every automorphism of the graph that preserves this
edge-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 are
Sylow cyclic and whose order is not divisible by four. We also provide several
new constructions of non-CCA graphs.
