Let $G$ be a group and $X$ a set. For an action $\mu\colon G\times X\longrightarrow X$, we can build a group morphism $\rho\colon G\longrightarrow Aut_{SET}(X)$ given by $\rho(g)(x)=\mu(g,x)$. In fact you can start with $\rho$ and arrive to the definition of $\mu$.
Let $G$ be a topological group and $X$ a topological space. A continuous action is asked to be an action $\mu$ such that is continuous with the product topology over $G\times X$. Does this implies that the induced $\rho$ is continuous when we take the open-compact topology over $Aut_{TOP}(X)$? If we start with the continuity of $\rho$, do we get the continuity of $\mu$?