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In this note, we prove that for every ordinary genus-2 curve $X$ over afinite field $\kappa$ of characteristic 2 with$\text{Aut}(X/\kappa)=\db{Z}/2\db{Z} \times S_3$, there exist$\text{SL}(2,\kappa\sembrack{s})$-representations of $\pi_1(X)$ such that theimage of $\pi_1(\bar{X})$ is infinite. This result gives a geometricinterpretation of Laszlo's counterexample [12] to a question regarding thefiniteness of the geometric monodromy of representations of the fundamentalgroup [4].

The Fixed Point Locus of the Verschiebung on M_x(2,0) for Genus-2 Curves X in Charateristic 2