Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion

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