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.



Residual properties of iterated semidirect products
The group law of $G^{\rtimes 2}$ is given by
$$(g, h)(g',h') = (ghg'h^{-1}, hh')$$ and more generally for $G^{\rtimes k}$, we have 
$(g\_1, \dots, g\_{k -1}, g\_k)^{(1, \dots,1, g)} = (g\_1^g, \dots,g\_{k - 1}^g, g\_k^{g})$ where $x^y \Doteq y^{-1}xy$.


**Therefore $G^{\rtimes k}$ is just $G^k$ in disguise**.


Indeed, we have $G^{\rtimes 2} = (G \times \{1\}) \cdot D(G)$ with $D(G) = 
\left\{ (g, g^{-1}) \, \vert \, g \in G \right\} \lhd G^{\rtimes 2}$ and 
$[G \times \{1\}, D(G)] \subset (G \times \{1\}) \cap D(G) = 1$.


Considering the following normal subgroups of $G^{\rtimes 3}$,
$$
\left\{ ((g, 1), 1) \, \vert \, g \in G \right\}, \,
\left\{ ((g, g^{-1}),1) \, \vert \, g \in G \right\},\,
\left\{ ((1, g), g^{-1}) \, \vert \, g \in G \right\}
$$
we see how the isomorphism $G^{\rtimes k} \simeq G^k$ generalizes for $k \ge 3$.


Turning back to OP's question, the two implications are then immediate.


