Let S be a surface of general type with not birational bicanonical map and that does not contain a pencil of genus 2 curves.If K 2 S = 8, p g (S) = 4 and q(S) = 0 then S can be given as double cover of a quadric surface.We show that its moduli space is generically smooth of dimension 38, and single out an open subset.Note that for these surfaces h 2 (S, T S ) is not zero.