  This paper develops a harmonic Galois theory for finite graphs, thereby
classifying harmonic branched $G$-covers of a fixed base $X$ in terms of
homomorphisms from a suitable fundamental group of $X$ together with
$G$-inertia structures on $X$. As applications, we show that finite embedding
problems for graphs have proper solutions and prove a Grunwald-Wang type result
stating that an arbitrary collection of fibers may be realized by a global
cover.
