Structure of Cross-wired Lamplighter Groups

We answer a question posed by Cornulier, Fisher, and Kashyap, proving that for a closed cocompact subgroup $\Gamma$ of $\text{Isom}(\text{DL}(n, n))$ not contained in $\text{Isom}^+(\text{DL}(n, n))$, the sequence $1 \to H \to \Gamma \to D_\infty \to 1$ splits, where $H$ is the unique open normal subgroup such that $\Gamma/H \cong D_\infty$.