  Three-manifold invariants $\hat Z$ (''$Z$-hat''), also known as homological
blocks, are $q$-series with integer coefficients. Explicit $q$-series form for
$\hat Z$ is known for $SU(2)$ group, supergroup $SU(2|1)$ and ortho-symplectic
supergroup $OSp(2|2)$. We focus on $\hat Z$ for $SO(3)$ group and
orthosymplectic supergroup $OSp(1|2)$ in this paper. Particularly, the change
of variable relating $SU(2)$ link invariants to the $SO(3)$ & $OSp(1|2)$ link
invariants plays a crucial role in explicitly writing the $q$-series.
