Stabilizers of consistent walks

A walk of length $n$ in a graph is consistent if there exists an automorphism of the graph that maps the initial $n-1$ vertices to the final $n-1$ vertices of the walk. In this paper we find some sufficient conditions for a consistent walk in an arc-transitive graph to have a trivial pointwise stabilizer. We show that in that case, the size of the smallest generating set of the group is bounded by the valence of the graph.