Matroids from gain graphs over quotient groups

We present a new construction for matroids from gain graphs that simultaneously generalizes several existing constructions. The construction takes as input a gain graph over a Frobenius group $\Gamma$ with Frobenius kernel $\Gamma_1$ and outputs an elementary lift of the frame matroid of the underlying gain graph over the quotient group $\Gamma/\Gamma_1$. While the hypothesis that $\Gamma$ is a Frobenius group may seem unusual, we prove that it is in some sense necessary: if $\Gamma$ is any finite group with a nontrivial proper normal subgroup $\Gamma_1$ and there is a construction that takes in a complete $\Gamma$-gain graph and outputs an elementary lift $M$ of the frame matroid of the underlying $(\Gamma/\Gamma_1)$-gain graph so that a cycle of the graph is a circuit of $M$ if and only if it is $\Gamma$-balanced, then $\Gamma$ is a Frobenius group with Frobenius kernel $\Gamma_1$.