  Starting with an O(2)-principal fibration over a closed oriented surface F_g,
g>=1, a 2-fold covering of the total space is said to be special when the
monodromy sends the fiber SO(2) = S^1 to the nontrivial element of Z_2.
Adapting D Jonhson's method [Spin structures and quadratic forms on surfaces, J
London Math Soc, 22 (1980) 365-373] we define an action of Sp(Z_2,2g), the
group of symplectic isomorphisms of (H_1(F_g;Z_2),.), on the set of special
2-fold coverings which has two orbits, one with 2^{g-1}(2^g+1) elements and one
with 2^{g-1}(2^g-1) elements. These two orbits are obtained by considering
Arf-invariants and some congruence of the derived matrices coming from Fox
Calculus. Sp(Z_2,2g) is described as the union of conjugacy classes of two
subgroups, each of them fixing a special 2-fold covering. Generators of these
two subgroups are made explicit.
