In the paper The ground axiom is consistent with $V\ne{\rm HOD}$ (J. D. Hamkins, J. Reitz, W.H. Woodin, PAMS 136(8):2008), we prove that the ground axiom is consistent with $V\neq\text{HOD}$, and remark there that:
The proof of Theorem 1 is
flexible and generalizes in a variety of ways. For example,
not much was used about the specific iteration $\mathbb{P}$. For
establishing GA, we needed to know only that the stage
$\gamma$ forcing was ${\lt}\gamma$-closed, and we could easily
have accommodated $\text{Add}(\gamma,\gamma^{++})$ or
occasionally $\text{Coll}(\gamma,\gamma^{+})$, for example,
without any difficulty in the argument. Also, we needn't
have forced specifically at every regular cardinal stage
$\gamma$, but could have forced at regular cardinals in
some other unbounded pattern. Thus, the argument
establishes that after forcing over $L$ with any of the
usual reverse Easton iterations of closed forcing, one
obtains the Ground Axiom in the extension.
Thus, with this kind of forcing, one can achieve a wide spectrum of patterns in the continuum function. Although the usual Easton forcing itself is a product, rather than an iteration, and will therefore definitely not result in the ground axiom, nevertheless one may transform the usual Easton product into an iteration, by taking big chunks of the forcing at a time. That is, given an Easton function $E$, one looks at sufficient closure points of $E$, and performs the iteration, which performs the usual Easton product between these closure points. The result will be a model of GA with the desired Easton function as the continuum function. I don't know if the details of this argument, however, have ever been written down, and if you were inclined in that direction, I would encourage you to do it (feel free to contact me).
