Structure of Cross-wired Lamplighter Groups

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