  This paper investigates some actions "\`a la Johnson" on the set, denoted by
${\cal E}$, of Spin-structures which are interpreted as special
double-coverings of a trivial $S^1-$fibration over a non-orientable surface
$N_{g+1}$. The group acting is first a group of orthogonal isomorphisms
assoiciated to $N_{g+1}$. A second approach is to consider the subspace of
${\cal E}$ (with $2^{g}$ elements) coming from special double-coverings of
$S^1\times F_g$, where $F_g$ is the orientation covering of $N_{g+1}$. The
group acting now is a subgroup of the group of symplectic isomorphisms
associated to $F_{g}$. In both situations, we obtain results on the number of
orbits and the number of elements in each orbit. Except in one case, these
results do not depend on any necessary choices. We compare both previous
classifications to a third one: weak-equivalence of coverings
