Relative sizes of iterated sumsets

Let $hA$ denote the $h$-fold sumset of a subset $A$ of an abelian group. Resolving a problem of Nathanson, we show that for any prescribed permutations ${\sigma }_1,\dots ,{\sigma }_H\in {\mathfrak{S}}_n$, there exist finite subsets $A_1,\dots ,A_n\subseteq \mathbb{Z}$ such that for each $1\le h\le H$, the relative order of the quantities $\mathrm{\mid }h{A}_1\mathrm{\mid },\dots ,\mathrm{\mid }h{A}_n\mathrm{\mid }$ is given by ${\sigma }_h$. We also establish extensions where $\mathbb{Z}$ is replaced by any other infinite abelian group or where one prescribes some equalities (not only inequalities) among the sumset sizes.