  We prove that the mod $2$ Betti numbers of double coverings of a complex
hyperplane arrangement complement are combinatorially determined. The proof is
based on a relation between the mod $2$ Aomoto complex and the transfer long
exact sequence.
  Applying the above result to the icosidodecahedral arrangement ($16$ planes
in the three dimensional space related to the icosidodecahedron), we conclude
that the first homology of the Milnor fiber has non-trivial $2$-torsion.
