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 ( 6 d ) and S 4 ( 6 c ). For each L ∈ { A 4 ( 6 ) , S 4 ( 6 d ) , S 4 ( 6 c ) }, we construct an infinite family of finite connected 6‐valent graphs { Γ n } n ∈ N and arc‐transitive groups G n ≤ Aut ( Γ 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 ∣ ( G n ) v ∣ is exponential in ∣ V ( Γ n ) ∣. 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.