Let $G$ be a topological group and $X$ be a topological space with $G$-action. Suppose there are two topologies $A$ and $B$ over $X$, with topology $A$ finer than $B$.
Question: Does $G$-action over $X$ being continuous (i.e. $\varphi: G\times X \to X$ being continuous) w.r.t topology $A$ imply it is also continuous w.r.t topology $B$?
For example is it true that: if $G$ acts continuous over $X$ for the discrete topology, then it is continuous for any other topology?
Remark.
1.For a given continuous map $f:X\to Y$, I know that it is still continuous when we replace topology of $X$ with a finer one and the topology of $Y$ with a coarser one. I didn't see why my question can be more subtle than this general case, but I have impression that this is actually true (but I can't come up with an easy argument, nor can I show this is wrong by some easy examples).
2.I know that when $X$ is endowed with the discrete topology, then action $G$ is continuous if the stabilizer $Stab_{G}(x)$ of any point $x\in X$ is open in $G$. I didn't see how can this help to show that $\varphi^{-1}(U)$ is open in $G \times X$ for any open $U$ (in another topology of $X$). I have $\varphi^{-1}(U)\supset \bigcup\limits_{x\in U}x\times Stab_{G}(x)$, but this is not enough to conclude.
Thanks for the nice counter-example given below by @Aryaman Maithani, the question hence has negative answer in general. I changed a little the question, cf. Continuity of Galois action under changing the topology of G-set to a coarser one.