Automorphisms of a Free Centre-by-Centre-by-Metabelian Group of Rank 3

Let $F_3$ be the free group of rank $3$ and let $G_3=F_3\mathrm{/}[F_3^{\mathrm{\prime }\mathrm{\prime }},F_3,F_3]$, that is, $G_3$ is a free centre-by-centre-by-metabelian group of rank $3$. We show that $\mathrm{A}\mathrm{u}\mathrm{t}(G_3)$ contains a proper finitely generated subgroup that is dense with respect to the formal power series topology.