Let $G$ be a Lie group whose lie algebra is $\mathfrak{g}$ with exponential map $exp:\mathfrak{g}\to G$.

For what kind of Lie group $G$ the standard process of definition of rotation number for circle homeomorphisms work well?

Namely for every homeomorphism $f:G\to G$ there is a homeomorphism $F:\mathfrak{g}\to \mathfrak{g}$ with $exp\circ F=f\circ exp$ and the limit of $$\frac{F^n(x)-x}{n}$$ as n goes to infinity exists?

This limit as an element of the Lie algebra would be called the rotation element.

What is a precise example for which this process work(other than $S^1$)?

Is it equivalents to the exponential to be a covering map?