  Magnetoelectric properties of a coupled spin-electron model on a doubly
decorated square lattice in an external electric field applied along the
crystallographic axis [11] are rigorously examined with the help of generalized
decoration-iteration transformation. The phase diagram, spontaneous
magnetization and electric polarization are exactly calculated and their
dependencies are comprehensively investigated under a concurrent influence of
temperature and electric field. It is found that the electric field mostly
stabilizes at zero temperature the spontaneous antiferromagnetic order with
respect to the ferromagnetic one. At finite temperatures the external electric
field gradually suppresses a spontaneous ferromagnetic (antiferromagnetic)
order emergent close to a quarter (half) filling. An enhanced magnetoelectric
response is detectable in vicinity of a continuous phase transition at which
the spontaneous magnetization vanishes and the electric polarization displays a
weak-type singularity. It is demonstrated that reentrant phase transitions of
the ferromagnetic or antiferromagnetic phase may be induced at moderate values
of the electric field, which simultaneously produces a sharp kink in a critical
line of the ferromagnetic phase nearby a quarter filling.
