Suppose we have a ring $R$ and a finite group $G$ acting on it, Is there a way to compute the invariant ring $R^G$ explicitly? Infact I am more interested in the case of affine ring and the symmetric group acting on it and I want to have the relations in the invariant ring explicitly.

More presicely, I have an algebra having finitely many generators and finitely many relation and the symmetric group acts on it. I want to have the relations in the invariant ring explicitly. I could not find a way to do so in the books I have seen so far.