Suppose we have a subgroup $H < \mathrm{Gl}(W)$, where $W = \mathbb{C}^n$, which acts on $\mathrm{End}(W)$ by precomposing: $a \mapsto h\circ a$, for $h\in H$.
We are given a distinguished linear subspace $L_0 \subset W$ and we consider the affine variety $$X = \bigcup_{h \in H} \mathrm{End}(W, h(L_0))$$ which contains the distinguished subvariety $Y = \mathrm{End}(W, L_0)$.
Problem: I would like to prove $X //H \simeq Y//H_0$ where $H_0$ is the stabiliser of $L_0$ (for the natural action of $H$ on $W$).
I am struggling to find a formal proof of this. Intuitively it should work, since for $h\in H$ we have $h\cdot Y=\mathrm{End}(W, h(L_0))$, so we can view $X = \bigcup_{h \in H} h\cdot Y$ and $H_0$ is the stabiliser of $Y$
but I'm a bit stuck and not so familiar with the details of GIT (if it has to do with that). Any suggestion?
Check the composition $Y\hookrightarrow X\twoheadrightarrow X/H$ is $H_0$-equivariant, which means there is a well-defined map $Y/H_0\to X/H$. Can you show this map is one-to-one and onto?