  We provide a theoretical model of $F(R)$ gravity in which it is possible to
describe in a unified way inflation, an early and a late dark energy era, in
the presence of a light axion particle which plays the role of the dark matter
component of the Universe. Particularly, the early-time phenomenology is
dominated by an $R^2$ term, while the presence of the other terms $f(R)$ ensure
the occurrence of the early and late-time dark energy eras. The inflationary
phenomenology is compatible with the Planck 2018 data for inflation, while the
late-time dark energy era is compatible with the Planck 2018 constraints on the
cosmological parameters. Also, the model exhibits an early dark energy era, at
$z\sim 2.5$ approximately, followed by a deceleration era, which starts at
approximately $z\sim 1.5$, which in turn is followed by a late-time dark energy
era for redshifts $z\sim 0.5$, which lasts for approximately 5 billion years up
to present time. A notable feature of our model is that the dark energy era is
free from dark energy oscillations, at least in the redshift interval
$z=[0,10]$. In addition, we also discuss several features related to
observational data at $z\sim 2.34$, at which redshift intricate observational
data exist in the literature. Moreover, the numerical code for the dark energy
phenomenology, written in Python 3, is presented in the end of the article.
Finally, the model has another interesting characteristic, a sudden jump of the
value of the Hubble rate in the redshift interval $z\sim [2,2.6]$ where its
value suddenly increases and then decreases until $z\sim 0$.
