  An extension of unimodular Einsteinian gravity in the context of $F(R)$
gravities is used to construct a class of anisotropic evolution scenarios. In
unimodular GR the determinant of the metric is constrained to be a fixed number
or a function. However, the metric of a generic anisotropic universe is not
compatible with the unimodular constraint, so that a redefinition of the
metric, to properly take into account the constraint, need be performed. The
unimodular constraint is imposed on $F(R)$ gravity in the Jordan frame by means
of a Lagrangian multiplier, to get the equations of motion. The resulting
equations can be viewed as a reconstruction method, which allows to determine
what function of the Ricci scalar can realize the desired evolution. For the
sake of clarity, some characteristic examples are invoked to show how this
reconstruction method works explicitly. The de Sitter spacetime here
considered, in the context of unimodular $F(R)$ gravity, is suitable to
describe both the early- and late-time epochs of the universe history.
