Let $\varphi:G\times X\to X$ be a continuous action such that $G$ be a finitely generated group, $X$ be a compact Hausdorff space, and $\mathcal{U}=\{U_i\}_{i=1}^m$ be a finite open cover of $X$.
Take finite open cover $\mathcal{V}=\{V_i\}_{i=1}^m$ with $\overline{V_i}\subseteq U_i$.
For every $n\in\mathbb{N}$, let $F_n\subseteq G$ be a finite set with $F_{n-1}\subseteq F_n$ and $G=\bigcup_{n\in\mathbb{N}}F_n$, also take $\Lambda_n\in \{1, 2, \ldots, m\}^{F_n}$.
For every $n\in\mathbb{N}$, take $x_n\in \bigcap_{g\in F_n} \varphi_g(V_{\Lambda_n(g)}$. Since $X$ is compact, hence we can assume that $x_n\to x$.
Is it true that $x\in \bigcap_{g\in G}\varphi_g(V_{\Lambda(g)})$?
where $\Lambda\in\{1, 2, \ldots, m\}^G$ define by $\Lambda(g)= \Lambda_n(g)$ if $g\in F_n$ but $g\notin F_{n-1}$.