In dynamical system $f:X\to X$, a point $x\in X$ is called a 2-periodic point if $f(x)\neq x$ anf $f^2(x)=x$. This implies that $x$ is a fixed point for $f^2:X\to X$.
Let $\varphi:G\times X\to X$ be a continuous action of a finitely generated group $G$ on compact metric space $X$. Let $K\subseteq X$ be a $G$-invariant finite subset of $X$ i.e. for every $g\in G$, $\varphi(g, K)=K$. If $K=\{x, \varphi(g_0, x), \varphi(g_1, x), \ldots, \varphi(g_n, x)\}$, is there a subgroup of finite index $H$ of $G$ such that $x$ is fixed point of $\varphi:H\times X\to X$? Please help me to know it.
This is true for any set theoretic action (and it has nothing to do with continuity nor with 2-periodic points, it is true for any $G$-invariant finite subset of $X$).
Since $K$ is $G$-invariant, by restriction there is an induced action $\phi : G \times K \to K$. One can think of this restricted action formally as a group homomorphism $\Phi : G \mapsto \text{Sym}(K)$, where $\text{Sym}(K)$ is the symmetric group of the set $K$, i.e. the group of self-bijections of $K$ under composition. But $K$ is a finite set, therefore $\text{Sym}(K)$ is a finite group, and therefore the image subgroup $\Phi(G) < \text{Sym}(K)$ is a finite group. It follows that the kernel $H = \text{Ker}(\Phi) < G$ is a finite index normal subgroup of $G$ (because the index of the kernel equals the cardinality of the image). Each element of $K$ is therefore a fixed point of each element of the finite index subgroup $H$.