A finite simple group is CCA if and only if it has no element of order four

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A Cayley graph for a group G is CCA if every automorphism of the graph that preserves the edge-orbits under the regular representation of G is an element of the normaliser of G. A group G is then said to be CCA if every connected Cayley graph on G is CCA. We show that a finite simple group is CCA if and only if it has no element of order 4. We also show that "many" 2-groups are non-CCA.