Conditions for Subjective Smooth Submersion to be Riemannian Submersion

Theorem
Let $\struct {\tilde M, \tilde g}$ and $\struct {M, g}$ be Riemannian manifolds.

Let $\pi : \tilde M \to M$ be a surjective smooth submersion.

Let $G$ be a group acting on $\tilde M$.

Suppose the action of $G$ is isometric, vertical, and transitive on fibers.

Then there exists a unique Riemannian metric $g$ such that $\pi$ is a Riemannian submersion.