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.
• Now I see that there are other examples for $\mathbb S^3$, thanks to Lee Mosher. It seems that taking joints you get such examples in higher dimensions