The goal of this article is to classify unramified covers of a fixed tropical base curve $Gamma$ with an action of a finite abelian group G that preserves and acts transitively on the fibers of the cover. We introduce the notion of dilated cohomology groups for a tropical curve $Gamma$, which generalize simplicial cohomology groups of $Gamma$ with coefficients in G by allowing nontrivial stabilizers at vertices and edges. We show that G-covers of $Gamma$ with a given collection of stabilizers are in natural bijection with the elements of the corresponding first dilated cohomology group of $Gamma$.
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