A question about tropical Pic(X) as a G-module

This paper addresses the following question: Let X be a tropical curve and let G be a finite subgroup of the automorphism group of X. Let D be a divisor and assume that its equivalence class [D] is G-invariant. Question: Is there always a D ′ ∈ [D] which is G-invariant? This was answered by Goldstein, Guralnick, and Joyner [GGJ] in the case of an irreducible algebraic curve over an algebraically closed field. We try to extend the arguments of [GGJ] to the tropical case. For instance, in the tropical case, we prove a tropical analog of Hilbert’s Theorem 90. For our main result, we show that the answer to the above question is “yes ” for all tropical curves.