Edge-transitive core-free Nest graphs

A finite simple graph $\Gamma$ is called a Nest graph if it is regular of valency $6$ and admits an automorphism $\rho$ with two orbits of the same length such that at least one of the subgraphs induced by these orbits is a cycle. We say that $\Gamma$ is core-free if no non-trivial subgroup of the group generated by $\rho$ is normal in $\mathrm{Aut}(\Gamma)$. In this paper, we show that, if $\Gamma$ is edge-transitive and core-free, then it is isomorphic to one of the following graphs: the complement of the Petersen graph, the Hamming graph $H(2,4)$, the Shrikhande graph and a certain normal $2$-cover of $K_{3,3}$ by $\mathbb{Z}_2^4$.