  With the successes of $f(R)$ theory as a neutral modification of the Einstein
general relativity (GR), we continue our study in this field and attempt to
find general natural and charged black hole (BH) solutions. In the previous
papers (arXiv:2012.05711 and arXiv:2010.04701), we applied the field equation
of the $f(R)$ gravity to a spherically symmetric space-time
$ds^2=-U(r)dt^2+\frac{dr^2}{V(r)}+r^2 \left( d\theta^2+\sin^2\theta d\phi^2
\right)$ with unequal metric potentials $U(r)$ and $V(r)$ and with/without
electric charge. To ensure the closed form of system of the resulting
differential equations, we assumed the derivative of the $f(R)$ with respect to
the scalar curvature $R$ to have a form $F_1(r)=\frac{df(R(r))}{dR(r)} \propto
\frac{c}{r^n}$ but in case $n>2$, the resulting black hole solutions
with/without charge do not generate asymptotically GR BH solutions in the limit
$c\rightarrow 0$ which means that the only case that can generate GR BHs is
$n=2$. In this paper, we assume another form, i.e., $F_1(r)=
1-\frac{F_0-\left(n-3\right)}{r^n}$ with a constant $F_0$ and show that we can
generate asymptotically GR BH solutions for $n>2$ but we show that $n=2$ case
is not allowed. This form of $F_1(r)$ could be the most acceptable physical
form that we can generate from it physical metric potentials that can have well
known asymptotic form and we obtain the metric of the Einstein general
relativity in the limit of $F_0\to n-3$. We show that the form of the electric
charge depends on $n$ and that $n\neq 2$. Our study shows that the power $n$ is
sensitive and why we should exclude the case $n=2$ for the choice of $F_1(r)$
presented in this study. We also study the physics of these black hole
solutions by calculating their thermodynamical quantities, like entropy, the
Hawking temperature and Gibb's free energy, and derive the stability conditions
by using geodesic deviations.
