  A group $G$ is complete group if it satisfies $Z(G)=e$ and $Aut(G)=Inn(G)$.
In this paper, on the one hand, we study the basic properties of generalized
Cayley graphs and characterize two classes isomorphic generalized generalized
Cayley graphs of complete groups.
  On the other hand, we give the sufficient and necessary conditions of
complete group to be $GCI$ group and restricted $GCI$ group. As an application,
we complete the classification of restricted $GCI$-groups for symmetric groups.
