The Ground Axiom ($GA$) is the assertion that the universe of sets ($V$) is not a forcing extension of any inner model $W$ by nontrivial forcing $P\in W$.

Is $GA$ consistent with any possible behavior of continuum function $\kappa\mapsto 2^{\kappa}$?

It seems in models of $GA$ like $L$ and some other canonical models the growth speed of continuum function is too low (e.g. $L\models GCH$). So the natural question is:

What is the consistency situation for faster growth speeds of $\kappa\mapsto 2^{\kappa}$?