Let $S$ be a topological space equipped with an action of finite group $G$. Let $κ$ be the quotient map. Take some $T⊂S$ such that $κ(T)=κ(S)$ and for each $g\in G$ either $gT=T$ or $gT∩T=∅$.
Take some $C\subset T$ such that $C$ is closed in $T$ and for each $g\in G$ either $gC=C$ or $gC\cap C=\varnothing.$
Moreover, suppose that $gC\cap C=\varnothing \Leftrightarrow gT\cap T=\varnothing.$
Is it true that $κ(C)$ is closed?
It does not seem so. Here is an example. Consider $n>1$ natural number and the space $X$ a union of segments $[i, i+1]$ where $i$ considered mod $n$ and thus $n+1 = 1$ ( it's the border of a regular polygon with $n$ sides). The group $G = \mathbb{Z}/n$ acts on $X$ by rotations. Let $T= [0,1)$. Note that $T$ is not closed in $X$. Take now $C = [1/2, 1) \subset T$, a closed subset of $T$. We have $Y \colon = X/G = S^1$ a circle ( a segment with identified ends). However, $\chi(C) = [1/2, 1)$, not closed in $Y$.