Folner sets in a quotient of a f.g. amenable group

Let $G$ be a finitely generated amenable group.
I know that it’s a basic result that every quotient of $G$ is amenable.

Is it also true that every Folner sequence of $G$ projects onto a Folner sequence of the quotient? Is there a direct proof of that?

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It behaves quite bad. For instance, consider the projection $\mathbf{Z}^2\to\mathbf{Z}$, $(m,n)\mapsto n$, and $F_n=\{1,\dots,n\}^2\cup (J_n\times\{0\})$, where $J_n$ is a subset of the set of even negative integers, of cardinal $\ll n^2$. Then $(F_n)$ is a Følner sequence but its projection is not.

It’s just that this is not the good point of view. The closest point of view is replacing almost invariant subsets with almost invariant probabilities (a probability on a discrete is just a non-negative $\ell^1$-function with sum 1), and the latter behaves well with push-forwards because it takes into account the cardinal of the fibers. In the previous case, if we pass from the Følner subsets to the associated uniform probabilities, (by normalizing the indicator function of $F_n$), the push-forward is no longer an indicator function and gives negligible weight to the parasite subset $J_n$.

In general, as above, passing from Følner subsets to almost invariant probabilities is trivial; while the converse is more involved but now classical.