My question is triggered by my confusion with the notation $\Psi$ in Constructive Approach to Automorphism Groups of Planar Graphs by Klavík et al.
The notation $\Psi$ was first used expressing general finite group while referring to Frucht’s Theorem on page 1. On page 2, Theorem 2 takes the special case where $\Psi$ is the subgroup of automorphism group of trees. The notation is $\Psi \in Aut(TREE)$ instead of $\Psi \le Aut(TREE)$ but I still consider the authors wanted to mean subgroups.
In Theorem 3, it is said that
$\Psi_i$ is a direct product of symmetric, cyclic and dihedral groups
It is not obvious whether $\Psi \in Aut(TREE)$ is still true.
From On automorphism groups of networks by MacArthur, I understand due to Polya that
automorphism groups of trees belong to the class of permutation groups which contains the symmetric groups and is closed under taking direct and wreath products.
MacArthur didn't mention cyclic and dihedral groups as Klavik did.
Should I still assume that $\Psi \in Aut(TREE)$ still held in the Theorem 3 of Klavik's paper?