  Let $\mathbb{F}$ be a field and let $G\subset \mathbb{F}\setminus \{0\}$ be a
multiplicative subgroup. We consider the category $\mathcal{Cob}_G$ of
$3$-dimensional cobordisms equipped with a representation of their fundamental
group in $G$, and the category ${Vect}_{\mathbb{F},\pm G}$ of
$\mathbb{F}$-linear maps defined up to multiplication by an element of $\pm G$.
Using the elementary theory of Reidemeister torsions, we construct a
"Reidemeister functor" from $Cob_G$ to ${Vect}_{\mathbb{F},\pm G}$. In
particular, when the group $G$ is free abelian and $\mathbb{F}$ is the field of
fractions of the group ring $\mathbb{Z}[G]$, we obtain a functorial formulation
of an Alexander-type invariant introduced by Lescop for $3$-manifolds with
boundary; when $G$ is trivial, the Reidemeister functor specializes to the TQFT
developed by Frohman and Nicas to enclose the Alexander polynomial of knots.
The study of the Reidemeister functor is carried out for any multiplicative
subgroup $G\subset \mathbb{F}\setminus \{0\}$. We obtain a duality result and
we show that the resulting projective representation of the monoid of homology
cobordisms is equivalent to the Magnus representation combined with the
relative Reidemeister torsion.
