Automorphism group of the complete alternating group graph

Let $S_n$ and $A_n$ denote the symmetric group and alternating group of degree $n$ with $n\ge 3$, respectively. Let $S$ be the set of all $3$-cycles in $S_n$. The \emph{complete alternating group graph}, denoted by $CA{G}_n$, is defined as the Cayley graph $\mathrm{C}\mathrm{a}\mathrm{y}(A_n,S)$ on $A_n$ with respect to $S$. In this paper, we show that $CA{G}_n$ ($n\ge 4$) is not a normal Cayley graph. Furthermore, the automorphism group of $CA{G}_n$ for $n\ge 5$ is obtained, which equals to $$\mathrm{A}\mathrm{u}\mathrm{t}$(CAG_n)=(R(A_n)\rtimes $\mathrm{I}\mathrm{n}\mathrm{n}$(S_n))\rtimes $\mathbb{Z}$_2\cong (A_n\rtimes S_n)\rtimes $\mathbb{Z}$_2$,where$R(A_n)$ is the right regular representation of $A_n$, $\mathrm{I}\mathrm{n}\mathrm{n}(S_n)$ is the inner automorphism group of $S_n$, and ${\mathbb{Z}}_2=⟨h⟩$, where $h$ is the map $\alpha \mapsto {\alpha }^{-1}$ ($\mathrm{\forall }\alpha \in A_n$).