  In this paper we investigate, from the dynamical systems perspective, the
evolution of a Kantowski-Sachs metric in a generic class of $f(R)$ models. We
present conditions (i. e., differentiability conditions, existence of minima,
monotony intervals, etc.) for a free input function related to the $f(R)$, that
guarantee the asymptotic stability of well-motivated physical solutions,
specially, self-accelerated solutions, allowing to describe both inflationary-
and late-time acceleration stages of the cosmic evolution. We discuss which
$f(R)$ theories allows for a cosmic evolution with an acceptable matter era, in
correspondence to the modern cosmological paradigm. We find a very rich
behavior, and amongst others the universe can result in isotropized solutions
with observables in agreement with observations, such as de Sitter,
quintessence-like, or phantom solutions. Additionally, we find that a
cosmological bounce and turnaround are realized in a part of the
parameter-space as a consequence of the metric choice.
