New lift matroids for group-labeled graphs.

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.