What is known about isometric actions on $\mathbb S^n$ such that the quotient space is homeomorphic to $\mathbb S^n$?

**Comments.**

- I am mostly interested in (maybe trivial) properties of such actions for large $n$.
- I see that the orientation preserving part of Coxeter's group has this property.
- Originally I thought that any such action is generated by rotations around $\mathbb S^{n-2}$'s; now I see that there are other examples for $\mathbb S^3$; thanks to Lee Mosher.