  Invariants of 3-manifolds from a non semi-simple category of modules over a
version of quantum sl(2) were obtained by the last three authors in
[arXiv:1404.7289]. In their construction the quantum parameter $q$ is a root of
unity of order $2r$ where $r>1$ is odd or congruent to $2$ modulo $4$. In this
paper we consider the remaining cases where $r$ is congruent to zero modulo $4$
and produce invariants of $3$-manifolds with colored links, equipped with
generalized spin structure. For a given $3$-manifold $M$, the relevant
generalized spin structures are (non canonically) parametrized by
$H^1(M;\mathbb C/2\mathbb Z)$.
