Consider $G^{\rtimes k} := ((G \rtimes \dots ) \rtimes G)\rtimes G$ with diagonal action by inner automorphisms, $G^{\rtimes 1} = G$. Let $\mathcal P$ be a collection of groups. Is it true that

- $G$ residually $\mathcal P$ $\Rightarrow $ $G^{\rtimes \geq 2}$ residually $\mathcal P$? (very unlikely, but I can't find a counterexample)
- $G^{\rtimes 2}$ residually $\mathcal P$ $\Rightarrow $ $G^{\rtimes \geq 2}$ residually $\mathcal P$?

I'm mostly interested in first question with $\mathcal P$ a quasivariety and $G$ finitely generated, but anything goes.