  The generalized decoration-iteration transformation is adapted for the exact
study of a coupled spin-electron model on 2D lattices in which localized Ising
spins reside on nodal lattice sites and mobile electrons are delocalized over
pairs of decorating sites. The model takes into account a hopping term for
mobile electrons, the Ising coupling between mobile electrons and localized
spins as well as the Ising coupling between localized spins ($J'$). The ground
state, spontaneous magnetization and specific heat are examined for both
ferromagnetic ($J'>0$) as well as antiferromagnetic ($J'<0$) interaction
between the localized spins. Several kinds of reentrant transitions between the
paramagnetic ($P$), antiferromagnetic ($AF$) and ferromagnetic ($F$) phases
have been found either with a single critical point, or with two consecutive
critical points ($P-AF$/$F-P$) and three successive critical points
$AF/F-P-F/AF-P$. Striking thermal variations of the spontaneous magnetization
depict a strong reduction due to the interplay between annealed disorder and
quantum fluctuations in addition to the aforementioned reentrance. It is shown
that the specific heat displays diverse thermal dependencies including finite
cusps at the critical temperatures.
