  Let $G$ be a graph whose edges are labeled by a group. Zaslavsky's lift
matroid $M$ of this group-labeled graph is an elementary lift of the graphic
matroid $M(G)$ that respects the group-labeling; specifically, the cycles of
$G$ that are circuits of $M$ coincide with the cycles that are balanced with
respect to the group-labeling. For $k \ge 2$, when does there exist a rank-$k$
lift of $M(G)$ that respects the group-labeling in this same sense? For abelian
groups, we show that such a matroid exists if and only if the group is
isomorphic to the additive group of a non-prime finite field.
