  Let $X$ be an abstract tropical curve and let $G$ be a finite subgroup of the
automorphism group of $X$. Let $D$ be a divisor on $X$ whose equivalence class
is $G$-invariant. We address the following question: is there always a divisor
$D'$ in the equivalence class of $D$ which is $G$-invariant? Our main result is
that the answer is "yes" for all abstract tropical curves. A key step in our
proof is a tropical analogue of Hilbert's Theorem 90.
