Is there a transitive set of disjoint diagonals of an icosidodecahedron?

Question

An icosidodecahedron has thirty vertices, upon which the symmetry group of the polyhedron acts transitively. Is there a set of fifteen non-intersecting diagonals of the icosidodecahedron (i.e., line segments that are not edges but both of whose endpoints are vertices) such that the symmetry group of the icosidodecahedron acts transitively on these diagonals? You can see the motivation for the question and some failed attempts here.