  We investigate the generation of helical magnetic fields and address the
baryon asymmetry of the universe from an inflationary magnetogenesis scenario,
in which the conformal and parity symmetries of the electromagnetic field are
broken through its coupling to the Ricci scalar and to the Gauss-Bonnet
invariant via the dual field tensor, so that the generated magnetic field can
have a helical nature. Depending on the reheating mechanism, we consider two
different cases - (1) instantaneous reheating scenario, in which case the
reheating e-fold number is zero, and (2) Kamionkowski reheating scenario which
is parametrized by a non-zero e-fold number, a reheating equation of state
parameter and a given reheating temperature. It is demonstrated that for both
the reheating mechanisms, the generated magnetic fields can be compatible with
the observations for suitable range of the model parameter present in the
non-minimal coupling of the electromagnetic field. Actually the present
magnetogenesis model does not produce sufficient hierarchy between the electric
and magnetic fields at the end of inflation, and thus the electric field is not
able to sufficiently induce (or enhance) the magnetic field during the
Kamionkowski reheating stage. This in turn makes both the reheating cases
almost similar from the perspective of magnetic field's evolution. Furthermore
we find that the magnetic fields at the galactic scale with strength $\sim
10^{-13}\mathrm{G}$ can lead to the resultant value of the ratio of the
baryonic number density to the entropy density as large as $\sim 10^{-10}$,
which is consistent with the observational data.
