Classification of tetravalent $2$-transitive non-normal Cayley graphs of finite simple groups

A graph $\mathrm{\Gamma }$ is called $(G,s)$-arc-transitive if $G \le $\mathrm{A}\mathrm{u}\mathrm{t}$(\Gamma)$istransitiveonthesetofverticesof$\Gamma$ and the set of $s$-arcs of $\mathrm{\Gamma }$, where for an integer $s\ge 1$ an $s$-arc of $\mathrm{\Gamma }$ is a sequence of $s+1$ vertices $(v_0,v_1,\dots ,v_s)$ of $\mathrm{\Gamma }$ such that $v_{i-1}$ and $v_i$ are adjacent for $1\le i\le s$ and $v_{i-1}\ne v_{i+1}$for$1 \le i \le s-1$\mathrm{.}$\Gamma$iscalled2-transitiveifitis$($\mathrm{A}\mathrm{u}\mathrm{t}$(\Gamma), 2)$-arc-transitivebutnot$($\mathrm{A}\mathrm{u}\mathrm{t}$(\Gamma), 3)$-arc-transitive\mathrm{.}ACayleygraph$\Gamma$ofagroup$G$iscallednormalif$G$isnormalin$$\mathrm{A}\mathrm{u}\mathrm{t}$(\Gamma)$ and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if $\mathrm{\Gamma }$ is a tetravalent 2-transitive Cayley graph of a finite simple group $G$, then either $\mathrm{\Gamma }$ is normal or $G$ is one of the groups ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(11)$, $M_{11}$, $M_{23}$ and $A_{11}$. However, it was unknown whether $\mathrm{\Gamma }$ is normal when $G$ is one of these four groups. In the present paper we answer this question by proving that among these four groups only $M_{11}$ produces connected tetravalent 2-transitive non-normal Cayley graphs. We prove further that there are exactly two such graphs which are non-isomorphic and both determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.