Three local actions in $6$-valent arc-transitive graphs

It is known that there are precisely three transitive permutation groups of degree $6$ that admit an invariant partition with three parts of size $2$ such that the kernel of the action on the parts has order $4$; these groups are called $A_4(6)$, $S_4(6d)$ and $S_4(6c)$. For each $L\in {A_4(6), S_4(6d), S_4(6c)}$,weconstructaninfinitefamilyoffiniteconnected$6$-valent graphs $\{{\mathrm{\Gamma }}_n{\}}_{n\in \mathbb{N}}$ and arc-transitive groups $G_n \le \rm{Aut}(\Gamma_n)$ such that the permutation group induced by the action of the vertex-stabiliser $(G_n{)}_v$ on the neighbourhood of a vertex $v$ is permutation isomorphic to $L$, and such that $\mathrm{\mid }(G_n{)}_v\mathrm{\mid }$ is exponential in $\mathrm{\mid }\mathrm{V}({\mathrm{\Gamma }}_n)\mathrm{\mid }$. These three groups were the only transitive permutation groups of degree at most $7$ for which the existence of such a family was undecided. In the process, we construct an infinite family of cubic $2$-arc-transitive graphs such that the dimension of the $1$-eigenspace over the field of order $2$ of the adjacency matrix of the graph grows linearly with the order of the graph.