I have asked this question on StackExchange, but haven't got any reply.

https://math.stackexchange.com/questions/2729335/is-the-quotient-of-a-toric-variety-by-a-finite-group-still-toric

Suppose $X$ is a toric variety with fan $\Sigma$, and the lattice of one-parameter subgroups of its torus is $N$. Suppose $\phi$ is a representation of finite group $G$ in $N \otimes_{\mathbb{Z}} \mathbb{R}$, \begin{equation} \phi:G \rightarrow \text{GL}( N \otimes_{\mathbb{Z}} \mathbb{R}) \end{equation} which sends the lattice $N$ to $N$, and the cones of $\Sigma$ to cones of $\Sigma$, therefore each $\phi(g)$ defines a toric automorphism of $X$.

Question 1: is the quotient variety $X/G$ (which exists from GIT) still a toric variety? Is the quotient morphism $X \rightarrow X/G$ toric?

Question 2: if so, how to construct the fan of $X/G$ and the toric morphism $X \rightarrow X/G$?

Any references on the two questions?

I have asked this question on Math.StackExchange, but haven't got any reply.

Suppose $X$ is a toric variety with fan $\Sigma$, and the lattice of one-parameter subgroups of its torus is $N$. Suppose $\phi$ is a representation of finite group $G$ in $N \otimes_{\mathbb{Z}} \mathbb{R}$, \begin{equation} \phi:G \rightarrow \text{GL}( N \otimes_{\mathbb{Z}} \mathbb{R}) \end{equation} which sends the lattice $N$ to $N$, and the cones of $\Sigma$ to cones of $\Sigma$, therefore each $\phi(g)$ defines a toric automorphism of $X$.

Question 1: Is the quotient variety $X/G$ (which exists from GIT) still a toric variety? Is the quotient morphism $X \rightarrow X/G$ toric?

Question 2: If so, how to construct the fan of $X/G$ and the toric morphism $X \rightarrow X/G$?

Any references on the two questions?