Some bidouble planes with $p_g=q=0$ and $4\le K^2\le 7$

We give a list of possibilities for surfaces of general type with $p_g=0$ having an involution $i$ such that the bicanonical map of $S$ is not composed with $i$ and $S\mathrm{/}i$ is not rational. Some examples with $K^2=4,\mathrm{.}\mathrm{.}\mathrm{.},7$ are constructed as double coverings of an Enriques surface. These surfaces have a description as bidouble coverings of the plane.