  In this paper we investigate graphs that admit a group acting
arc-transitively such that the local action is semiprimitive with a regular
normal nilpotent subgroup. This type of semiprimitive group is a generalisation
of an affine group. We show that if the graph has valency coprime to six, then
there is a bound on the order of the vertex stabilisers depending on the
valency alone. We also prove a detailed structure theorem for the vertex
stabilisers in the remaining case. This is a contribution to an ongoing project
to investigate the validity of the Poto\v{c}nik-Spiga-Verret Conjecture.
