  In a series of papers we proposed a model unifying general relativity and
quantum mechanics. The idea was to deduce both general relativity and quantum
mechanics from a noncommutative algebra ${\cal A}_{\Gamma}$ defined on a
transformation groupoid $\Gamma$ determined by the action of the Lorentz group
on the frame bundle $(E, \pi_M, M)$ over space-time $M$. In the present work,
we construct a simplified version of the gravitational sector of this model in
which the Lorentz group is replaced by a finite group $G$ and the frame bundle
is trivial $E=M\times G$. The model is fully computable. We define the
Einstein-Hilbert action, with the help of which we derive the generalized
vacuum Einstein equations. When the equations are projected to space-time
(giving the "general relativistic limit"), the extra terms that appear due to
our generalization can be interpreted as "matter terms", as in
Kaluza-Klein-type models. To illustrate this effect we further simplify the
metric matrix to a block diagonal form, compute for it the generalized Einstein
equations and find two of their "Friedmann-like" solutions for the special case
when $G =\mathbb{Z}_2$. One of them gives the flat Minkowski space-time (which,
however, is not static), another, a hyperbolic, linearly expanding universe.
