Local systems and Suzuki groups

We study geometric monodromy groups $G_{\text{geo},{\text{sF}}_q}$ of the local systems ${\text{sF}}_q$ on the affine line over ${\text{F}}_2$ of rank $D=\sqrt{q}(q-1)$, $q=2^{2n+1}$, constructed in \cite{Ka-ERS}. The main result of the paper shows that $G_{\text{geo},{\text{sF}}_q}$ is either the Suzuki simple group $\text{tw}2B_2(q)$, or the special linear group ${\text{SL}}_D$. We also show that ${\text{sF}}_8$ has geometric monodromy group $\text{tw}2B_2(8)$, and arithmetic monodromy group $\text{Aut}(\text{tw}2B_2(8))$ over ${\text{F}}_2$, thus establishing \cite[Conjecture 2.2]{Ka-ERS} in full in the case $q=8$.