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.