  An interesting fact is that most of the known connected $2$-arc-transitive
nonnormal Cayley graphs of small valency on finite simple groups are
$(\mathrm{A}_{n+1},2)$-arc-transitive Cayley graphs on $\mathrm{A}_n$. This
motivates the study of $2$-arc-transitive Cayley graphs on $\mathrm{A}_n$ for
arbitrary valency. In this paper, we characterize the automorphism groups of
such graphs. In particular, we show that for a non-complete
$(G,2)$-arc-transitive Cayley graph on $\mathrm{A}_n$ with $G$ almost simple,
the socle of $G$ is either $\mathrm{A}_{n+1}$ or $\mathrm{A}_{n+2}$. We also
construct the first infinite family of $(\mathrm{A}_{n+2},2)$-arc-transitive
Cayley graphs on $\mathrm{A}_n$.
