  We construct three dimensional non-semisimple topological field theories from
the unrolled quantum group of the Lie superalgebra $\mathfrak{osp}(1 \vert 2)$.
More precisely, the quantum group depends on a root of unity $q=e^{\frac{2 \pi
\sqrt{-1}}{r}}$, where $r$ is a positive integer greater than $2$, and the
construction applies when $r$ is not congruent to $4$ modulo $8$. The algebraic
result which underlies the construction is the existence of a relative modular
structure on the non-finite, non-semisimple category of weight modules for the
quantum group. We prove a Verlinde formula which allows for the computation of
dimensions and Euler characteristics of topological field theory state spaces
of unmarked surfaces. When $r$ is congruent to $\pm 1$ or $\pm 2$ modulo $8$,
we relate the resulting $3$-manifold invariants with physicists'
$\widehat{Z}$-invariants associated to $\mathfrak{osp}(1 \vert 2)$. Finally, we
establish a relation between $\widehat{Z}$-invariants associated to
$\mathfrak{sl}(2)$ and $\mathfrak{osp}(1 \vert 2)$ which was conjectured in the
physics literature.
