Let $(M, g)$ be a Riemannian manifold, and let $G$ be a group acting freely and properly on $M$. From differential geometry we know that the quotient set $M/G$, i.e., the set of the orbits, is a differential manifold and the quotient map $\pi:M \to M/G$ is a submersion.
My question is the following: there is a natural/general way for transferring the metric $g$ to $M/G$, in the two cases: (i) G acts isometrically or (ii) not.