Let $G$ be an amenable group and let $(H_i)_{i\in I}$ be the poset of all the countable (or, if simpler, even finitely generated) subgroups of $G$ so, in particular, we have $G=\bigcup_IH_i$. Furthermore, each $H_i$ is amenable (as it is a subgroup of an amenable group) and countable, so it has a Følner sequence $(F_{n,i})_{n\in\mathbb N}$. In this situation, is it possible to see that $(F_{n,i})_{(n,i)\in \mathbb N\times I}$ (with $\mathbb N\times I$ endowed with the product order) is a Følner net for $G$?
If this is not true in general, is it possible to choose the Følner sequences $(F_{n,i})_{n\in\mathbb N}$ in some special way that makes the result true?
Finally, if also my second question has a negative answer, is there some other canonical way to build a Følner net for $G$ out of Følner sequences (or nets) for the $H_i$'s?