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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 composedwith $i$ and $S/i$ is not rational. Some examples with $K^2=4,...,7$ areconstructed as double coverings of an Enriques surface. These surfaces have adescription as bidouble coverings of the plane.

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