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

• I am mostly interested in (maybe trivial) properties of such actions for large $n$. Say, is it is true that any such action is generated by rotations around $\mathbb S^{n-2}$'s; what else?
• 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.
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# Actions on ��ⁿSⁿ with quotient ≅𝕊ⁿSⁿ

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

• I am mostly interested in (maybe trivial) properties of such actions for large $n$. Say, is it true 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. It seems that taking joints you get such examples in higher dimensions