Suppose $X$ is a toric variety with fan $\Sigma$, and the lattice of its torus is $N$. Suppose there is a representation of finite group $G$,
\begin{equation}
\phi:G \rightarrow \text{GL}(N \otimes_{\mathbb{Z}} \mathbb{R})
\end{equation}
which maps the lattice points of $N$ to lattice points of $N$, and maps the cones of $\Sigma$ to the cones of $\Sigma$.
Question: is the quotient variety $X/G$ (which exists from GIT) still a toric variety?
I guess the answer is yes? Are there references about how to construct the fan of $X/G$?
Edit: I have asked this question on MO.