Bi-primitive 2-arc-transitive bi-Cayley graphs

A bipartite graph \(\Gamma \) $\mathrm{\Gamma }$ is a bi-Cayley graph over a group H if \(H\leqslant \textrm{Aut}\Gamma \) $H⩽\text{Aut}\mathrm{\Gamma }$ acts regularly on each part of \(\Gamma \) $\mathrm{\Gamma }$ . A bi-Cayley graph \(\Gamma \) $\mathrm{\Gamma }$ is said to be a normal bi-Cayley graph over H if \(H\unlhd \textrm{Aut}\Gamma \) $H⊴\text{Aut}\mathrm{\Gamma }$ , and bi-primitive if the bipartition preserving subgroup of \(\textrm{Aut}\Gamma \) $\text{Aut}\mathrm{\Gamma }$ acts primitively on each part of \(\Gamma \) $\mathrm{\Gamma }$ . In this paper, a classification is given for 2-arc-transitive bi-Cayley graphs which are bi-primitive and non-normal.