On the structure of the group of equivariant diffeomorphisms of G-manifolds with codimension one orbit

It is known that the equivariant diffeomorphism group DiffG∞(M)0 of a principal G-manifold M is perfect. If M has at least two orbit types, then it is not true. The purpose of this paper is to determine the first homology group of DiffG∞(M)0 when M is a G-manifold with codimension one orbit.