  Dimer tree algebras are a class of non-commutative Gorenstein algebras of
Gorenstein dimension 1. In previous work we showed that the stable category of
Cohen-Macaulay modules of a dimer tree algebra $A$ is a 2-cluster category of
Dynkin type $\mathbb{A}$. Here we show that, if $A$ has an admissible action by
the group $G$ with two elements, then the stable Cohen-Macaulay category of the
skew 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 terms
of a punctured polygon $\mathcal{P}$ with a checkerboard pattern in its
interior, such that the 2-arcs in $\mathcal{P}$ correspond to indecomposable
syzygies in $AG$ and 2-pivots correspond to morphisms. In particular, the dimer
tree algebras and their skew group algebras are Gorenstein algebras of finite
Cohen-Macaulay type $\mathbb{A}$ and $\mathbb{D}$ respectively. We also provide
examples of types $\mathbb{E}_6,\mathbb{E}_7,$ and $\mathbb{E}_8$.
