  For a finite group action on a finite EI quiver, we construct its `orbifold'
quotient EI quiver. The free EI category associated to the quotient EI quiver
is equivalent to the skew group category with respect to the given group
action. Specializing the result to a finite group action on a finite acyclic
quiver, we prove that, under reasonable conditions, the skew group category of
the path category is equivalent to a finite EI category of Cartan type. If the
ground field is of characteristic $p$ and the acting group is a cyclic
$p$-group, we prove that the skew group algebra of the path algebra is Morita
equivalent to the algebra associated to a Cartan matrix, defined in [C. Geiss,
B. Leclerc, and J. Schr\"{o}er, Quivers with relations for symmetrizable Cartan
matrices I: Foundations, Invent. Math. 209 (2017), 61--158]. We apply the
Morita equivalence to construct a categorification of the folding projection
between the root lattices with respect to a graph automorphism. In the Dynkin
cases, the restriction of the categorification to indecomposable modules
corresponds to the folding of positive roots.
