G-endomorphisms of affine G-varieties which induce automorphisms of the invariant subrings of the coordinate rings

Let G be a complex algebraic group and let X=SpecR be an affine G-variety. Let φ be a G-endomorphism of X which induces an automorphism of the invariant subring RG. We show that φ is a G-automorphism when G is unipotent and X is factorial. When G is reductive, we give some conditions under which φ is a G-automorphism.