Let $X=G/H$, where $G$ is a connected linear algebraic group over the field $\mathbf{C}$ of complex numbers and $H\subset G$ is an algebraic subgroup. In general, we can write the algebraic variety $X$ as $X=G'/H'$ with $G'$ non-isomorphic to $G$. What can be said about $G'$ (and $H'$)?

Of course, we can always take any surjective homomorphism $G'\to G$, then $X$ is naturally a homogeneous space of $G'$. Conversely, if the action of $G$ on $X$ is not effective, we can take a normal subgroup $N$ of $G$, contained in $H$ (say, the kernel of the action) and set $G'=G/N,\ H'=H/N$, then $X=G'/H'$. I am interested in less trivial examples.

I would be happy to get a general answer, but will be also grateful for special cases, examples, comments, etc.

I am obliged to Günter Harder for this nice title of my question.