Given a rational elliptic surface $R \to \Bbb P^1$, is there a way to know if there exists a supersingular K3 surface that arises as a base curve change $S=R\times_{\Bbb P^1} \Bbb P^1 \to \Bbb P^1$, given by a degree-two morphism $f:\Bbb P^1 \to \Bbb P^1$? I'm aware that, depending on the ramification points of $f$, $S$ is going to be a K3 surface, from the singular fibers of $R$, we know the singular fibers of $R$ and the maximum sublattice from the Néron-Severi group of $S$ coming from the Néron-Severi group of R has rank 20 (together with the zero section and the class of a smooth fiber), thus a supersingular K3 obtained by this method will have two more sections, right? Does the dependence on the characteristics of the supersingularity implies that there exist no generic geometric construction like that yielding always supersingular K3?

On the other way around, given a supersingular K3, is there a way to know if there exists a rational elliptic surface under such K3?