  We present a model of $F(R)$ gravity in the presence of a string theory
motivated misalignment axion like particle materialized in terms of a canonical
scalar field minimally coupled with gravity, and we study the cosmological
phenomenology of the model, emphasizing mainly on the late-time era. The main
result of the paper is that inflation and the dark energy era may be realized
in a geometric way by an $F(R)$ gravity, while the axion is the dark matter
constituent of the Universe. The $F(R)$ gravity model consists of an $R^2$
term, which as we show dominates the evolution during the early time, thus
producing a viable inflationary phenomenology, and a power law term $\sim
R^{\delta}$ with $\delta\ll 1 $ and positive, which eventually controls the
late-time era. The axion field remains frozen during the inflationary era,
which is an effect known for misalignment axions, but as the Universe expands,
the axion starts to oscillate, and its energy density scales eventually as we
show, as $\rho_a\sim a^{-3}$. After appropriately rewriting the gravitational
equations in terms of the redshift $z$, we study in detail the late-time
phenomenology of the model, and we compare the results with the $\Lambda$CDM
model and the latest Planck 2018 data. As we show, the model for small
redshifts $0<z<5$ is phenomenologically similar to the $\Lambda$CDM model,
however at large redshifts and deeply in the matter domination era, the results
are different from those of the $\Lambda$CDM model due to the dark energy
oscillations. For the late-time study we investigate the behavior of several
well-known statefinder quantities, like the deceleration parameter, the jerk
and $Om(z)$, and we demonstrate that the statefinders which contain lower
derivatives of the Hubble rate have similar behavior for both the $\Lambda$CDM
and the axion $F(R)$ gravity model.
