The fundamental groupoid functor $\pi$ does not commutes with colimits, in particular with quotients : take $I=[-1,1]$, quotiented by $\mathbb Z/2 \mathbb Z$ : you simply get an interval, but the quotient of $\pi I$ by $\mathbb Z/2 \mathbb Z$ is equivalent to $\mathsf{B} (\mathbb Z / 2 \mathbb Z)$.
I know that $\pi$ commutes with the quotient map when the group is discrete and acts freely, see https://doi.org/10.1016/S0001-8708(03)00072-0 by Emmanuel Dror Farjoun. This is because $\pi$ commutes with homotopy colimits.
However, I wondered if there is a reference for when the quotient is by a free action of a topological group; To be clear, I am asking if $\pi(X/G) = \pi X/\pi G$. In particular, this would hold if such a quotient $X/G$ is a homotopy colimit, and $\pi X / \pi G$ is a homotopy colimit for the standard model structure of the category of groupoids.