  We introduce admissible group actions on cluster algebras, cluster categories
and quivers with potential and study the resulting orbit spaces. The orbit
space of the cluster algebra has the structure of a generalized cluster
algebra. This generalized cluster structure is different from those introduced
by Chekhov-Shapiro and Lam-Pylyavskyy. For group actions on cluster algebras
from surfaces, we describe the generalized cluster structure of the orbit space
in terms of a triangulated orbifold. In this case, we give a complete list of
exchange polynomials, and we classify the algebras of rank 1 and 2. We also
show that every admissible group action on a cluster category induces a
precovering from the cluster category to the cluster category of orbits.
Moreover this precovering is dense if the categories are of finite type.
