* Suppose with have a topological manifold $X$ and a group $G$, is there a way to compute the fundamental group of $X/G$ in function of $\pi(X)$ and $\pi(G)$?
* are there any settings on X that can simplify the computing of $\pi(X/G)$ in function of $\pi(X)$ and $\pi(G)$?


could you please suggest to me a reference where can I find related topics?



fundamental group of $X/\mathbb{R}^n$
See Proposition 8.10 in Chapter I of [Transformation Groups and Algebraic K-Theory](https://link.springer.com/book/10.1007/BFb0083681) by Lück. When a Lie group $G$ acts properly on a connected manifold $X$, the proposition provides an exact sequence
$$
\pi\_1(X,x\_0)\rightarrow\pi\_1(X/G,x\_0G)\rightarrow\pi\_0(G)/M\rightarrow 0
$$
where $M$ is the normal subgroup of $\pi\_0(G)$ generated by $\pi\_0$ of the stabilizers. A description of the kernel of the first morphism $\pi\_1(X,x\_0)\rightarrow\pi\_1(X/G,x\_0G)$ can also be found there.


