Genus-two quartic from a degree-four complete linear system

On a smooth projective curve of genus two, $D=K+P+Q$ has three sections and no base points by the Riemann-Roch theorem. Its map to ${\mathbb{P}}^2$ is nondegenerate and has mapping degree times image degree four. The only nonbirational possibility is a double cover of a conic. Every degree-two divisor with two sections is canonical, again by Riemann-Roch theorem, so that possibility forces $D\sim 2K$, or $P+Q\sim K$. If this equivalence fails, the image is a quartic and the map is birational. The quartic is singular because a smooth plane quartic has genus three.