Is there $T_2$- space $X$ and a group $G$ such that $X_G$ is not $T_2$ space?

Question

Let $(X, \tau)$ be a topological space and the group $G$ acts on $X$ by $\varphi:G\times X\to X$. Let us define equivalence relation $\sim$ on $X$ by $x\sim y$ whenever there is $g\in G$ such that $y=\varphi(g,x)$. Let $X_G$ be quotient space with respect to $\pi:X\to X_G$.

Is there $T_2$- space $X$ and a group $G$ such that $X_G$ is not $T_2$ space?

Answer 1

Yes. Take $X=\mathbb R$, endowed with the usual topology, and $G=(\mathbb{Q},+)$ again endowed with the usual topology. Let $\varphi(g,x)=g+x$. Then $\mathbb{R}_{\mathbb Q}$ is not $T_2$.

Answer 2

If $G$ is compact Hausdorff, then no (see Theorem 3.1 in Bredon's Introduction to compact transformation groups). Otherwise, even for $G=({\mathbf R},+)$ and $X$ compact Hausdorff, this is not true.

Consider $X={\bf R}^2/{\bf Z}^2$ and the action via $r\cdot (a,b)=(a+r,b+r\sqrt 2)$. Then every orbit is dense, so $X_G$ has trivial topology.