  For a complex variety $\hat X$ with an action of a reductive group $\hat G$
and a geometric quotient $\pi: \hat X \to X$ by a closed normal subgroup $H
\subset \hat G$, we show that open sets of $X$ admitting good quotients by
$G=\hat G / H$ correspond bijectively to open sets in $\hat X$ with good $\hat
G$-quotients. We use this to compute GIT-chambers and their associated
quotients for the diagonal action of $\text{PGL}_2$ on $(\mathbb{P}^1)^n$ in
certain subcones of the $\text{PGL}_2$-effective cone via a torus action on
affine space. This allows us to represent these quotients as toric varieties
with fans determined by convex geometry.
