Fundamental domains for rhombic lattices with dihedral symmetry of order 8

We show by construction that every rhombic lattice $\mathrm{\Gamma }$ in ${\mathbb{R}}^2$ has a fundamental domain whose symmetry group contains the point group of $\mathrm{\Gamma }$ as a subgroup of index $2$. This solves the last open case of a question raised in [3] on fundamental domains for planar lattices whose symmetry groups properly contain the point groups of the lattices.