Suppose that $G$ is a reductive group that acts algebraically on an affine variety $X$ over an algebraically closed field $k$.

Suppose also that $G$ is equipped with a normal abelian subgroup $N$. such that the quotient group $G/N$ is finite abelian.

Example: $G = \text{O}(2,k)$ and $N= \text{SO}(2,k)$, and $k = \Bbb C$ is the complex numbers.

My question: Under what circumstances can I conclude that a categorical/good quotient $X/\!\!/G$ coincides with a categorical/good quotient for $G/N$ acting on $X/\!\!/N$?

That is, under what circumstances is there a birational equivalence between $X/\!\!/G$ and $$ (X/\!\!/N)/\!\!/(G/N) \,\, ? $$ If there is such a result, can someone point me to a place in the literature where it can be found?

Suppose that $G$ is a reductive group that acts algebraically on an affine variety $X$ over an algebraically closed field $k$.

Suppose also that $G$ is equipped with a normal abelian subgroup $N$ such that the quotient group $G/N$ is finite abelian.

Example: $G = \text{O}(2,k)$ and $N= \text{SO}(2,k)$, and $k = \Bbb C$ is the complex numbers.

My question: Under what circumstances can I conclude that a categorical/good quotient $X/\!\!/G$ coincides with a categorical/good quotient for $G/N$ acting on $X/\!\!/N$?

That is, under what circumstances is there a birational equivalence between $X/\!\!/G$ and $$ (X/\!\!/N)/\!\!/(G/N) \,\, ? $$ If there is such a result, can someone point me to a place in the literature where it can be found?

Suppose that $G$ is a reductive group that acts algebraically on an affine variety $X$ over an algebraically closed field $k$.

Suppose also that $G$ is equipped with a maximal torus $T$ such that the quotient group $G/T$ is finite abelian.

My question: Under what circumstances can I conclude that a categorical/good quotient $X/\!\!/G$ coincides with a categorical/good quotient for $G/T$ acting on $X/\!\!/T$?

That is, under what circumstances is there a birational equivalence between $X/\!\!/G$ and $$ (X/\!\!/T)/\!\!/(G/T) \,\, ? $$ If there is such a result, can someone point me to a place in the literature where it can be found?

Suppose that $G$ is a reductive group that acts algebraically on an affine variety $X$ over an algebraically closed field $k$.

Suppose also that $G$ is equipped with a normal abelian subgroup $N$. such that the quotient group $G/N$ is finite abelian.

Example: $G = \text{O}(2,k)$ and $N= \text{SO}(2,k)$, and $k = \Bbb C$ is the complex numbers.

My question: Under what circumstances can I conclude that a categorical/good quotient $X/\!\!/G$ coincides with a categorical/good quotient for $G/N$ acting on $X/\!\!/N$?

That is, under what circumstances is there a birational equivalence between $X/\!\!/G$ and $$ (X/\!\!/N)/\!\!/(G/N) \,\, ? $$ If there is such a result, can someone point me to a place in the literature where it can be found?