  In this work, we shall incorporate a stiff era in the Universe's evolution in
the context of $F(R)$ gravity. After deriving the vacuum $F(R)$ gravity, which
may realize a stiff evolution, we combine the stiff $F(R)$ gravity with an
$R^2$ model, and we construct a qualitative model for the inflationary and
stiff era, with the latter commencing after the end of the inflationary era. We
assume that the baryogenesis occurs during the stiff era, and we calculate the
baryon to entropy ratio, which effectively constraints the functional form of
the stiff $F(R)$ gravity. Further constraints on the stiff $F(R)$ gravity may
come from the primordial gravitational waves, and particularly their scalar
mode, which is characteristic of the $F(R)$ gravity theory. The stiff era
presence does not contradict the standard cosmology era, namely, inflation, and
the radiation-matter domination eras. Furthermore, we investigate which $F(R)$
gravity may realize a dust and stiff matter dominated Einstein-Hilbert
evolution.
