We prove that if $G$ is a subgroup of the (time-change) symmetry group of a Markov process $X_t$ which is transitive and has a compact isotropy subgroup, then after a time change, $X_t$ becomes $G$-invariant. The symmetry groups of diffusion processes are discussed in more detail. We show that if the generator of $X_t$ is the Laplacian with respect to the intrinsic metric, then $X_t$ has the best invariance property.