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Statements:

1) The automorphisms of the colored Cayley digraph of group $G$, is equal to the group $G$ itself. I assumed this is true based on the proof sketch of Frutch's theorem.

2) But then consider the automorphisms of a cycle of length N. This cycle is the Cayley graph of the cyclic group $C_N$. However its automorphism group is the Dihadral group $D_N$.

Which one of my statements is wrong?

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    Neither of them is wrong. As a graph, we have a cycle of length N and its automorphism group is dihedral. On the other hand, as a colored Cayley digraph on a cyclic group, you have to replace edges by arcs in each direction, and those arcs have different colors (inverses of each other), so that "reflection" is no longer an automorphism. – 2017-02-01
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    Seems so obvious now! Thanks, that was killing me. – 2017-02-01

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