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

Abstract: 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 $S/i$ is not rational. Some examples with $K^2=4,...,7$ are constructed as double coverings of an Enriques surface. These surfaces have a description as bidouble coverings of the plane.