I'm looking for results of the form "every infinite solvable group contains <...> as a subgroup". Specifically, I believe:

If $G$ is infinite solvable, finitely generated and not virtually cyclic, then $G$ contains as subgroup either the metabelian group $\mathbb Z[1/mn]\rtimes\mathbb Z$, with $\mathbb Z$ acting by multiplication by $m/n$ [possibly $m=n=1$], or the wreath product $\mathbb Z/p\wr\mathbb Z$ for some prime $p$.

(I know that every infinite solvable group contains one of these two subgroups as a section [= quotient of two subgroups], by Kropholler's 1984 result; but really need them as subgroups. In fact, if the answer to the question is "no", I'm very much interested in a corrected statement with as few extra cases as possible.)

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