A finite simple graph $Γ$ is called a Nest graph if it is regular of valency $6$ and admits an automorphism $ρ$ 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 $Γ$ is core-free if no non-trivial subgroup of the group generated by $ρ$ is normal in $\mathrm{Aut}(Γ)$. In this paper, we show that, if $Γ$ 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 $\mathbb{Z}_2^4$.