  We propose a $SU(5) \times U(1)_X \times U(1)_{PQ}$ model, where $U(1)_X$ is
the generalization of the $B-L$ (baryon minus lepton number) gauge symmetry and
$U(1)_{PQ}$ is the global Peccei-Quinn (PQ) symmetry. There are four fermions
families in $\bf{{\overline 5}} + \bf{10}$ representations of $SU(5)$, a mirror
family in $\bf{5}+\bf{{\overline {10}}}$ representations, and three $SU(5)$
singlet Majorana fermions. The $U(1)_X$ related anomalies all cancel in the
presence of the Majorana neutrinos. The $SU(5)$ symmetry is broken at $M_{GUT}
\simeq (6-9)\times 10^{15}$ GeV and the proton lifetime $\tau_p$ is estimated
to be well within the expected sensitivity of the future Hyper-Kamiokande
experiment, $\tau_p \lesssim 1.3 \times 10^{35}$ years. The $SU(5)$ breaking
also triggers the breaking of the PQ symmetry, resulting in axion dark matter
(DM), with the axion decay constant $f_a$ of order $M_{GUT}$ or somewhat
larger. The CASPEr experiment can search for such an axion DM candidate. The
Hubble parameter during inflation must be low, $H_{inf} \lesssim 10^9 $ GeV, in
order to successfully resolve the axion domain wall, axion DM isocurvature and
$SU(5)$ monopole problems. With the identification of the $U(1)_X$ breaking
Higgs field with the inflaton field, we implement inflection-point inflation,
which is capable of realizing the desired value for $H_{inf}$. The vectorlike
fermions in the model are essential for achieving successful unification of the
SM gauge couplings as well as the phenomenological viability of both axion DM
and inflation scenario.
