$\pi_{\mathcal{D}}^{G_i}:G_i^{\sigma}\to
G_i^{\mathcal{D}}$, $i=1,2$ satisfy $$\pi_{\mathcal{D}}^{G_2}\circ\chi^{\sigma}=\chi^{\mathcal{D}}\circ \pi_{\mathcal{D}}^{G_1}$$

\[cor:room\] Let $\chi \colon G_1\to G_2$ be a surjective homomorphism between two metrizable groups $G_1$ and $G_2$. If $\sigma$ and $\alpha$ are cardinal numbers with $m(\sigma)\leq \alpha$ and $\alpha^\omega\leq \sigma$, then it is possible to find an independent ${G_{\delta}}$-dense subset $D$ of $G_1^\sigma$ satisfying the properties of Proposition \[room\] such that in addition $\chi^\sigma(D)$ is an independent subset of $G_2^\sigma$.

It suffices to repeat the proof of Lemma \[room\] taking care to choose the sets $F_N$ in such a way that $\chi^{\cup_{\gamma \in
  N}\mathcal{A}_\gamma}(F_N)$ is also independent.

\[prop:freetorsion\] Let $\chi \colon G\to {{\mathbb T\,}}$ be a surjective character of a compact metrizable group $G$. If $\sigma $ and $\alpha$ are cardinal numbers with $m(\sigma)\leq \alpha$, and $\alpha^\omega\leq \sigma$, then the topological group $G^\sigma$ contains an independent ${G_{\delta}}$-dense subset $F$ of cardinality $\alpha$ such that $F$ and $\chi^\sigma (F)$ generate isomorphic groups with property ${{\mathbf \sharp}}$.

We begin with a ${G_{\delta}}$-dense subset of $G^\sigma$, $D=\left\{
d_\eta \colon \eta<\alpha\right\}$, with the properties of Lemma \[room\] and Corollary \[cor:room\]. We have thus two families of sets $\{\mathcal{S}_\theta,\: \colon\: \theta\in[\alpha]^{\omega}\}$, $\{\mathcal{N}_\eta,\: \colon\: \eta<\alpha\}\subset\sigma$ with the properties (1) through (4) of that Lemma.

Next, for every $\theta\in [\alpha]^{\omega} $, 