  Let $\mathbb{K}$ be a field of characteristic $p$ and $G$ be a cyclic
$p$-group which acts on a finite acyclic quiver $Q$. The folding process
associates a Cartan triple to the action. We establish a Morita equivalence
between the skew group algebra of the preprojective algebra of $Q$ and the
generalized preprojective algebra associated to the Cartan triple in the sense
of Geiss, Leclerc and Schr\"{o}er. The Morita equivalence induces an
isomorphism between certain ideal monoids of these preprojective algebras,
which is compatible with the embedding of Weyl groups appearing in the folding
process.
