The Krasner–Kaloujnine universal embedding theorem states that any group extension of a group $H$ by a group $A$ is isomorphic to a subgroup of the regular wreath product $A \operatorname{Wr} H$. When studying finitely generated infinite groups (e.g. in geometric group theory), this statement seems to be of limited use as the regular wreath product $A \operatorname{Wr} H$ is infinitely generated.

I was curious to know under what conditions the regular wreath product $A \operatorname{Wr} H$ could be replaced by the restricted wreath product $A \operatorname{wr} H$, which is finitely generated whenever $A$ and $H$ are. I assume this is not always the case?

In particular, under what conditions is a finitely generated solvable group isomorphic to a subgroup of an iterated wreath product of abelian groups?