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

We classify all n-dimensional reduced Cohen-Macaulay modular quotient varieties (Formula presented.) and study their singularities, where p is a prime number and (Formula presented.) denotes the cyclic group of order 2p. In particular, we present an example that demonstrates that the problem proposed by Yasuda has a negative answer if the condition that “G is a small subgroup” was dropped.