The canonical module and anti-invariant elements

Let $S$ be a ring having canonical module; let $\mathfrak G$ be a finite group of automorphisms of this ring, and let $R$ be the subring of elements of $S$ invariant with respect to the action of $\mathfrak G$. We study the problem of existence and characterization of the canonical module of the ring $R$. In particular we apply our results to the problem of descent of the Gorenstein property of a ring.
Bibliography: 19 titles.