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?