Suppose I have a Gorenstein variety $X$ over $\mathbb{C}$ with trivial canonical bundle, and the action of a finite group $G$ on $X$, which acts trivially on the canonical bundle. When does the quotient $X/G$ have a crepant resolution? 

I have some very specific examples in mind, namely when $X$ is one of the cyclic covers of $\mathbb{P}^n$ considered by Sheng-Xu-Zuo in https://arxiv.org/pdf/1211.3646.pdf, which admits (many) crepant resolutions, and $G$ being another cyclic group; I am also not sure whether the action of $G$ extends to any resolution $\tilde{X}$ of $X$, though I am leaning towards no, which is why the question is phrased as above. On the other hand I would also like to know what the general expectation/state of the art is for the general question, so any help would be much appreciated!