The following are hints:
(1) Consider the case $G=H$ and consider one action of $G$ on $X$ to be defined by $g\cdot x=x$ for all $x\in X$. (Prove that this is indeed an action of $G$ on $X$ if you have not done so already.) The quotient space $X/G$ is naturally homeomorphic to $X$. Find an example of a non-trivial topological group $X$ and an action of $G$ on $X$ such that $X/G$ has exactly one orbit (for example). Conclude that $X/G$ is not homeomorphic to $X/H$.
(2) Let $k\in K$ be such that $k^{-1}Gk=H$. Prove that $x\mapsto k\cdot x$ induces a homeomorphism $X/G\to X/H$. (In particular, you need to prove that if two elements of $X$ lie in the same $G$-orbit, then their images under this map lie in the same $H$-orbit. For this, use $k^{-1}Gk=H$.)
(3) Consider two non-isomorphic groups acting trivially on $X$.