  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.
