Let $S/R$ be an extension of rings where $R$ is a domain, and let $G$ be a finite group acting on $S$, fixing $R$. When do we have $S^G \subset \text{Frac}(R)$?
For instance, if $S$ is a domain, $G$ acts on $\text{Frac}(S)$, and $\text{Frac}(S) / \text{Frac}(R)$ is Galois, then $S^G \subset \text{Frac}(S)^G \subset \text{Frac}(R)$.