  We obtain a type of Ay\'{o}n-Beato-Garc\'{\i}a (ABG) related black hole
solutions with five parameters: the mass $m$, the charge $q$, and three
dimensionless parameters $\alpha$, $\beta$ and $\gamma$ associated with
nonlinear electrodynamics. We find that this type of black holes is regular
under the conditions: $\alpha \gamma \geqslant 6$, $\beta \gamma \geqslant 8$,
and $\gamma >0$. Here we focus on the saturated case: $\alpha={6}/{\gamma}$ and
$\beta ={8}/{\gamma }$, such that only three parameters $m$, $q$ and $\gamma$
remain, which leads to a new family of ABG black holes. For such a family of
black holes, we investigate the influence of the charge $q$ and the parameter
$\gamma$ on the horizon radius and the Hawking temperature. In addition, we
calculate the quasinormal mode frequencies of massless scalar field
perturbations by using the sixth-order WKB approximation method and the
unstable circular null geodesic method in the eikonal limit. We also compute
the shadow radius for the new family of ABG black holes and use the shadow data
of the $M87^{*}$ black hole detected by the Event Horizon Telescope to provide
an upper limit on the charge $q$ of the new black holes. Using the shadow data
of the $M87^{*}$ black hole, we find that the upper limit of the charge $q$
increases rapidly at first and then slowly but does not exceed the mass of the
$M87^{*}$ black hole at last when the parameter $\gamma$ is increasing and
going to infinity, and that the data restrict the frequency range of the
fundamental mode with $l=1$ to $1.4\times 10^{-6}Hz\sim 1.9\times 10^{-6} Hz$.
