  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. On the one hand,
our results show that the increase of the charge $q$ makes the scalar waves
decay faster at first and then slowly except for the case of $\gamma=2$. On the
other hand, they show that the increase of the parameter $\gamma$ makes the
scalar waves decay at first sharply and then slowly. In particular, $\gamma=1$
can be regarded as the critical value for the transition from an unstable
configuration to a stable one. Finally, we compute the shadow radius for the
new family of ABG black holes. We find that the increase of the charge $q$
makes the shadow radius decrease monotonically, while the increase of the
parameter $\gamma$ makes the shadow radius increase at first rapidly and then
almost remain unchanged, especially the parameter $\gamma$ has a significant
impact on the shadow radius when it is less than one.
