Arc-transitive pentavalent Cayley graphs with soluble vertex stabilizer on finite nonabelian simple groups

A Cayley graph $\text{Ga}=\text{Cay}(G,S)$ is said to be normal if $G$ is normal in $\text{Aut}\text{Ga}$. The concept of normal Cayley graphs was first proposed by M.Y.Xu in [Discrete Math. 182, 309-319, 1998] and it plays an important role in determining the full automorphism groups of Cayley graphs. In this paper, we investigate the normality problem of the connected arc-transitive pentavalent Cayley graphs with soluble vertex stabilizer on finite nonabelian simple groups. We prove that all such graphs $\text{Ga}$ are either normal or $G={\text{A}}_{39}$ or ${\text{A}}_{79}$. Further, a connected arc-transitive pentavalent Cayley graph on ${\text{A}}_{79}$ is constructed. To our knowledge, this is the first known example of pentavalent 3-arc-transitive Cayley graph on finite nonabelian simple group which is non-normal.