gr.group theory - Quotient of manifolds by groups and embeddings

Let $f:X_1to X_2$ be a closed submanifold. Let $rho:G_1to G_2$ be a closed Lie subgroup. Let $G_1$ acts on $X_1$ and $G_2$ on $X_2$ and suppose $f$ is $rho$-equivariant. I would like to get a morphism $overline{f}:X_1/G_1to X_2/G_2$ and conditions for $overline{f}$ to be also a closed embedding. Do you know a reference where I can find this situation treated, in the category of analytic manifolds(or algebraic varieties)?