  We study Kuperberg invariants for sutured manifolds in the case of a
semidirect product of an involutory Hopf superalgebra $H$ with its automorphism
group $\text{Aut}(H)$. These are topological invariants of balanced sutured
3-manifolds endowed with a homomorphism of the fundamental group into
$\text{Aut}(H)$ and possibly with a $\text{Spin}^c$ structure and a homology
orientation. We show that these invariants are computed via a form of Fox
calculus and that, if $H$ is $\mathbb{N}$-graded, they can be extended in a
canonical way to polynomial invariants. When $H$ is an exterior algebra, we
show that this invariant specializes to a refinement of the twisted relative
Reidemeister torsion of sutured 3-manifolds. We also give an explanation of our
Fox calculus formulas in terms of a particular Hopf group-algebra.
