  Clifford theory relates the representation theory of finite groups to those
of a fixed normal subgroup by means of induction and restriction, which is an
adjoint pair of functors. We generalize this result to the situation of a
Krull-Schmidt category on which a finite group acts as automorphisms. This then
provides the orbit category introduced by Cibils and Marcos, and studied
intensively by Keller in the context of cluster algebras, and by Asashiba in
the context of Galois covering functors. We formulate and prove Clifford's
theorem for Krull-Schmidt orbit categories with respect to a finite group
$\Gamma$ of automorphisms, clarifying this way how the image of an
indecomposable object in the original category decomposes in the orbit
category. The pair of adjoint functors appears as the Kleisli category of the
naturally appearing monad given by $\Gamma$.
