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Anna Erschler proved (the paper referred to in the question) that for every group $G$ with Foelner function $F$ the Foelner function of $G\wr G$ is $F^F$. This implies that if $G$ is SQ-universal in the class of amenable groups, its Foelner function $F$ must satisfy $F\equiv F^F$. I do not remember exactly but that probably means that one cannot prove the amenability of $G$ (at least existence of the Foelner sets) using Peano arithmetic. Perhaps a more detailed analysis of what happens to the Foelner function under taking subgroups and homomorphic images would immediately imply that $G\wr G$ cannot embed into a factor-group of a subgroup of $G$ if $G$ is amenable. You can also ask Anna directly.

Update Simon has answered the original question. But still I think it is interesting to find out if there exists a finitely generated amenable group $G$ such that $G\wr G$ embeds to a homomorphic image of $G$. If such a $G$ exists its Foelner function must be truly remarkable.

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Anna Erschler proved (Erschler, Anna On isoperimetric profiles of finitely generated groups. Geom. Dedicata 100 (2003), 157–171.the paper referred to in the question) that for every group $G$ with Foelner function $F$ the Foelner function of $G\wr G$ is $F^F$. This implies that if $G$ is SQ-universal in the class of amenable groups, its Foelner function $F$ must satisfy $F\equiv F^F$. I do not remember exactly but that probably means that one cannot prove the amenability of $G$ (at least existence of the Foelner sets) using Peano arithmetic. Perhaps a more detailed analysis of what happens to the Foelner function under taking subgroups and homomorphic images would immediately imply that $G\wr G$ cannot embed into a factor-group of a subgroup of $G$ if $G$ is amenable. You can also ask Anna directly.

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Anna Erschler proved (Erschler, Anna On isoperimetric profiles of finitely generated groups. Geom. Dedicata 100 (2003), 157–171.) that for every group $G$ with Foelner function $F$ the Foelner function of $G\wr G$ is $F^F$. This implies that if $G$ is SQ-universal in the class of amenable groups, its Foelner function $F$ must satisfy $F\equiv F^F$. I do not remember exactly but that probably means that one cannot prove the amenability of $G$ (at least existence of the Foelner sets) usin using Peano arithmetic. Perhaps a more detailed analysis of what happens to the Foelner function under taking subgroups and homomorphic images would immediately imply that $G\wr G$ cannot embed into a factor-group of a subgroup of $G$ if $G$ is amenable. You can also ask Anna directly.

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