Let $X$ be a topological space and let $G$ be a topological group which acts on $X$ as $\theta:G\times X\to X$. I am interested in what conditions might be placed on $G$ and $X$ to make the following two definitions equivalent.
Def 1: Let us say that a group action $\theta:G\times X\to X$ is continuous if and only if the map $\theta:G\times X\to X$ is itself a continuous map.
In general, this implies but is not equivalent to the following condition:
Def 2: Let us say that a group action $\theta:G\times X\to X$ acts by homeomorphisms if and only if for each $g\in G$ the map $\theta_g:X\to X$ defined by $\theta_g(p):=\theta(g,p)$ is a homeomorphism, $\theta_g\in \text{Homeo}(X)$.
The second definition is automatically met when we consider the case of $H\subset\text{Homeo}(X)$ a topological group of homeomorphisms which acts by the natural group action $\theta_\text{nat}:H\times X\to X$ with $\theta_\text{nat}(h,x)=h(x)$. The first definition, however, might not be met by this action. But intuitively a topological group of homeomorphisms "should" act continuously. How can I restrict $G$ and $X$ to make this so?
For instance, I know that if we make $X$ a smooth manifold and act by a Lie group of diffeomorphisms, $H\subset\text{Diff}(X)\subset\text{Homeo}(X)$, then these two conditions are equivalent. After a bit of work, I think this follows from the Lie group exponential being a continuous map. Can the same effect be achieved with weaker assumptions about $X$ and $H$?
Returning to the generic case, I also think that a topological group of homeomorphisms $H\subset\text{Homeo}(X)$ "should" act continuously on $X$ by virtue of the fact that they are homeomorphisms. But for this to be the case, $H$ would need to pick up its topology from $\text{Homeo}(X)$, but I read recently that $\text{Homeo}(X)$ itself isn't a topological group in general.