Orbits of actions of group superschemes

Working over an algebraically closed field k, we prove that all orbits of a left action of an algebraic group superscheme G on a superscheme X of finite type are locally closed. Moreover, such an orbit Gx, where x is a k-point of X, is closed if and only if G_evx is closed in X_ev, or equivalently, if and only if G_resx is closed in X_res. Here G_ev is the largest purely even group super-subscheme of G and G_res is G_ev regarded as a group scheme. Similarly, X_ev is the largest purely even super-subscheme of X and X_res is X_ev regarded as a scheme. We also prove that sdim(Gx)=sdim(G)−sdim(G_x), where G_x is the stabilizer of x.