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I am reading the answer here, but I don't understand why "the fibres look like cosets $H/gHg^{-1}$."

Here are my thoughts. Fix some $H$-orbit of $\tilde{X}$ and pick a point $x$ in the orbit. The fibre is precisely all the orbits that are hit by $g^{-1}hx$ for all $h\in H$. First, we observe that $g^{-1}(ghg^{-1})x=hg^{-1}x$, so in fact the actions of $ghg^{-1}$ and the identity $e$ on $x$ are sent to the same orbit. We now need to show that if $h$ sends $x$ to the same orbit as $e$, then $h=gh^*g^{-1}$ for some $h^*\in H$. Writing this down as an equation yields $h'g^{-1}x=g^{-1}hx$, or in other words $gh'g^{-1}x=hx$. But I'm not sure how to proceed from here, as all we know is that $gh'g^{-1}$ is off from $h$ by a multiple of the stabilizer.

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