Modular quotient varieties and singularities by the cyclic group of order $2p$

We classify all $n$-dimensional reduced Cohen-Macaulay modular quotient variety $mathbb{A}_mathbb{F}^n/C_{2p}$ and study their singularities, where $p$ is a prime number and $C_{2p}$ denotes the cyclic group of order $2p$. In particular, we present an example that demonstrates that the problem proposed by Yasuda cite[Problem 6.6]{Yas2015} has a negative answer if the condition that $G$ is a small subgroup was dropped.