Let $X,Y$ be $G$-spaces ($G$ a topological group). In general $X$ and $Y$ can be homeomorphic as topological spaces without being isomorphic as $G$-spaces.
For example $X=Y=S^1$ and $G=\mathbb{Z}_2$ acting on $X$ by rotating by $\pi$ and on $Y$ by reflecting along $x$-axis. Then $X$ and $Y$ are clearly homeomorphic but not isomorphic as $G$-spaces, since the action on $X$ has no fixed points and the action on $Y$ has two fixed points $-1$ and $1$.
What are sufficient conditions for $G$-spaces $X,Y$ so that homeomorphic $\iff$ isomorphic as $G$-spaces?
For example, clearly, if $X,Y$ are trivial $G$-spaces this equivalence holds.