rt.representation theory - A question on group action on algebras

Let $A = k[x,y]/(x^2,xy,y^2)$ and let the generator of the two-element group act by $x mapsto -x$, $y mapsto -y$.

More generally, take any finite group acting on $R = k[x_1, dots, x_n]$ with invariant subring $S = k[f_1, dots, f_m]$, and set $A = R/(f_1, dots, f_m)$.