Let $X$ be a set and $\mathfrak{S}_X$ the symmetric group of $X$. Let $G$ be a group and suppose we have a homomorphism $$\phi:G\rightarrow\mathfrak{S}_X.$$ For every $x\in X$, let $\omega_x:G\rightarrow E$ be defined by $\omega_x(g)=\phi_g(x)$, for all $g\in G$. Then for every $x\in X$ the mapping $${}^x\phi:G\rightarrow\mathfrak{S}_{\omega_x(G)},\,g\mapsto\phi_g|_{\omega_x(G)},$$ is a group homomorphism: i.e. $\omega_x(G)$ is a homogenous $G$-set.
What is the relation between the group $\text{Im}(\phi)$ and the family of groups $(\text{Im}({}^x\phi))_{x\in X}$? Is there a way to identify/embed $\text{Im}(\phi)$ with/into $\prod_{x\in X}\text{Im}({}^x\phi)$?