This is a follow up of this question. Given a toric variety $X$ with a fan $\Sigma$ and a finite group $G$ acting on $X$, we know that the GIT quotient $X/G$ exists. However, as stated in the answer to the linked question, the quotient need not be a toric variety.

Are there any conditions for $X$ or $G$ so that the quotient $X/G$ is a toric variety?

Any references are welcome. Thank you in advance.



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