Let $R$ = $K[X,Y]$, where $K = \overline{K}$, and char $K = 0$. Let $\mathfrak {m} = (X,Y)$. Let $A = R^H$, where $H$ is a finite group of automorphisms of $R$, and $\sigma (\mathfrak{m}) \subset \mathfrak {m} \, \forall \, \sigma \in H$. Let $\mathfrak{n} = A \cap \mathfrak {m}$.
Thus $R_{\mathfrak{m}}$ is a regular local ring of dimension $2$, $R_{\mathfrak{m}}$ dominates $A_{\mathfrak{n}}$ and from problem 12 of Chapter 5 of Atiyah-MacDonald, we have $R_{\mathfrak{m}}$ is integral over $A_{\mathfrak{n}}$.
My question is: can we show that $A_{\mathfrak{n}}$ is a regular local ring of dimension $2$ ?