Automorphisms of Cubic Surfaces and the Clebsch Map

It is a classical result that a non-singular cubic surface with 27 lines can be constructed from a non-conical 6-arc in the plane via a Clebsch Map.  We investigate the action of automorphisms of the cubic surface on the points in the preimage of the Clebsch map, with the aim of better understanding the relationship between cubic surfaces and the non-conical 6-arcs.