  We consider piecewise quadratic perturbations of centers of piecewise
quadratic systems in two zones determined by a straight line through the
origin. By means of expansions of the displacement map, we are able to find
isolated zeros of it, without dealing with the unsurprising difficult integrals
inherent in the usual averaging approach. We apply our technique to non-smooth
perturbations of the four families of isochronous centers of the Loud family,
$S_1$, $S_2$, $S_3$, and $S_4$, as well as to non-smooth perturbations of
non-smooth centers given by putting different $S_i$'s in each zone. To show the
coverage of our approach, we apply its first order, which recovers the
averaging method of the first order, in perturbations of the already mentioned
centers considering all the straight lines through the origin. Then we apply
its second order to perturbations of the above centers for a specific oblique
straight line. Here in order to argue we introduce certain blow-ups in the
perturbative parameters. As a consequence of our study, we obtain examples of
piecewise quadratic systems with at least $12$ limit cycles. By analyzing two
previous works of the literature claiming much more limit cycles we found some
mistakes in the calculations. Therefore, the best lower bound for the number of
limit cycles of a piecewise quadratic system is up to now the $12$ limit cycles
found in the present paper.
