  We describe an algorithm to compute the zeta function of any
non-hyperelliptic genus 3 plane curve $C$ over a finite field with automorphism
group $G = \mathbb{Z} / 2 \mathbb{Z}$. This algorithm computes in the
Monsky-Washnitzer cohomology of~the curve. Using the relation between the
Monsky-Washnitzer cohomology of $C$ and its quotient $E := C/G$, the
computation splits into 2 parts: one in a subspace of the Monsky-Washnitzer
cohomology and a second which reduces to the point counting on an elliptic
curve $E$. The former corresponds to the dimension $2$ abelian surface
$\mathrm{ker}(\mathrm{Jac}(C) \rightarrow E)$, on which we can compute with
lower precision and with matrices of smaller dimension. Hence we obtain a
faster algorithm than working directly on the curve $C$.
