Question

Let $Y$ be an affine algebraic variety over $\mathbb{C}$ and let $X$ be its closed subvariety. Let $G$ be a reductive group acting on $Y$ and let $H$ be a reductive subgroup of $G$ preserving $X$ such that the induced map $\phi: X//H \to Y//G$ is $1$-$1$ and a finite map. Question: Is $\phi$ a closed immersion?

Rmk. If the image of $\phi$ is normal then one can prove that the answer is affirmative.

Answer 1

Here is a counterexample. Let $Y = \mathbb A^2$, and let $G$ be a cyclic group of order $2$; a generator acts via $(x, y) \mapsto (-x, y)$. We take as $H$ the trivial subgroup, and as $X$ the curve $x^3= y$.