The classification of surfaces with $p_g=0$, $K^2=6$ and non birational bicanonical map

Let $S$ be a minimal surface of general type with $p_g=0$ and $K^2=6$, such that its bicanonical map $\text{fie}\text{\hspace{0.17em}⁣}:S\to {\text{pp}}^6$ is not birational. The map $\text{fie}$ is a morphism of degree $\le 4$ onto a surface. The case of $\mathrm{deg}\text{fie}=4$ is completely characterized in [Topology, {\bf 40} (5) (2001), 977–991] and the present paper completes the classification of these surfaces. It is proven that the degree of $\text{fie}$ cannot be equal to 3, and the geometry of surfaces with $\mathrm{deg}\text{fie}=2$ is analysed in detail. The last section contains three examples of such surfaces, two of which appear to be new.