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Dimer tree algebras are a class of non-commutative Gorenstein algebras ofGorenstein dimension 1. In previous work we showed that the stable category ofCohen-Macaulay modules of a dimer tree algebra $A$ is a 2-cluster category ofDynkin type $\mathbb{A}$. Here we show that, if $A$ has an admissible action bythe group $G$ with two elements, then the stable Cohen-Macaulay category of theskew group algebra $AG$ is a 2-cluster category of Dynkin type $\mathbb{D}$.This result is reminiscent of and inspired by a result by Reiten and Riedtmann,who showed that for an admissible $G$-action on the path algebra of type$\mathbb{A}$ the resulting skew group algebra is of type $\mathbb{D}$.Moreover, we provide a geometric model of the syzygy category of $AG$ in termsof a punctured polygon $\mathcal{P}$ with a checkerboard pattern in itsinterior, such that the 2-arcs in $\mathcal{P}$ correspond to indecomposablesyzygies in $AG$ and 2-pivots correspond to morphisms. In particular, the dimertree algebras and their skew group algebras are Gorenstein algebras of finiteCohen-Macaulay type $\mathbb{A}$ and $\mathbb{D}$ respectively. We also provideexamples of types $\mathbb{E}_6,\mathbb{E}_7,$ and $\mathbb{E}_8$.

On Gorenstein algebras of finite Cohen-Macaulay type: dimer tree algebras and their skew group algebras