We present in this paper a construction for Gabor-type frames built out of
generalized Weyl-Heisenberg groups. These latter are obtained via central
extensions of groups which are direct products of locally compact abelian
groups and their duals. Our results generalize many of the results, appearing
in the literature, on frames built out of the Schr\"odinger representation of
the standard Weyl-Heisenberg group. In particular, we obtain a generalization
of the result in \cite{PO}, in which the product $ab$ determines whether it is
possible for the Gabor system $\{E_{mb}T_{na}g \}_{m,n\in \mathbb Z}$ to be a
frame for $L^2(\mathbb R)$. As a particular example of the theory, we study in
some detail the case of the generalized Weyl-Heisenberg group built out of the
$d$-dimensional torus. In the same spirit we also construct generalized
shift-invariant systems.