Consider the the Powerset Size Axiom, that is, the following:

(PSA) ($\forall$x,y) |x|$\lt$|y|$\Rightarrow$$2^{|x|}$$\lt$$2^{|y|}$.

Does there exist a class $\mathscr M$ of models of ZFC such that the following holds:

ZFC+PSA$\vdash$"The Whitehead problem is answered in the Affirmative (i.e. Every Whitehead group is free)"

ZFC+$\lnot$PSA$\vdash$"The Whitehead Problem is answered in the Negative (i.e. there are nonfree Whitehead groups)"

If so, can a model of ZFC+$\lnot$PSA be a generic extension of a model of ZFC+PSA?