I have a couple of questions concerning existence and description of transfers for Bloch groups and scissors congruence groups/pre-Bloch groups.

To fix notation and recall definitions:

From the general algebraic K-theory machinery, we get transfers on $K_3$. In particular, for a finite field extension $E/F$, we get a map $\operatorname{tr}_{E/F}:K_3(E)\to K_3(F)$ such that the composition $K_3(F)\to K_3(E)\stackrel{\operatorname{tr}_{E/F}}{\longrightarrow}K_3(F)$ is multiplication with the degree $[E:F]$.

Now by the work of Bloch, Dupont-Sah, Suslin and others, we have another description of $K_3(F)^{\operatorname{ind}}=K_3(F)/K_3^M(F)$ in terms of an exact sequence $$ 0\to \widetilde{\operatorname{Tor}}(\mu(F),\mu(F))\to K_3(F)^{\operatorname{ind}}\to B(F)\to 0. $$ In the above, the Bloch group $B(F)$ is defined as $$ B(F)=\ker\left(\mathcal{P}(F)\to \Lambda^2(F^\times):[x]\to x\wedge(1-x)\right) $$ and the group $\mathcal{P}(F)$ is the pre-Bloch group or scissors congruence group $$ \mathcal{P}(F)=\left(\bigoplus_{x\in F^\times\setminus\{1\}}\mathbb{Z}[x]\right) /\left([x]-[y]+[y/x]-[(1-x^{-1})/(1-y^{-1})]+[(1-x)/(1-y)]\right), $$ the relation coming from the five-term relation satisfied by the dilogarithm. In particular, elements of the Bloch group $B(F)$ can be written down as linear combinations of symbols $[x], x\in F^\times\setminus\{1\}$ satisfying certain relations.

Now I can formulate my questions on transfers on Bloch groups and scissors congruence groups.

Does the transfer on $K_3$ induce a transfer on the Bloch group? I think that this is not the case in general. The element $[x]+[1-x]\in B(F)$ is independent of $x$, and is typically denoted by $c_F$. The element $c_{\mathbb{R}}$ has exact order $6$ in $B(\mathbb{R})$, and the element $c_{\mathbb{C}}$ is trivial in $B(\mathbb{C})$. This seems to contradict transfers for the Bloch group (it does not contradict transfers for $K_3^{\operatorname{ind}}$ because the torsion moves from $B(F)$ to $\widetilde{\operatorname{Tor}}(\mu(F),\mu(F))$). Are there more torsion elements like this, in particular with other odd orders? Are there further obstructions to the existence of transfers on Bloch groups? If $F$ contains an algebraically closed fields, it follows from work of Suslin and Levine that $B(F)$ is uniquely $\ell$-divisible for $\ell$ different from the characteristic - in particular $c_F=0$. Does the $K$-theory transfer induce a transfer on the Bloch group in this situation? What would be a good reference?

Is there an explicit description of what the transfer map on $K_3$ does on the Bloch group? I would be interested in a description that only uses the definition of the Bloch group via points on $\mathbb{P}^1(F)\setminus\{0,1,\infty\}$ given above.

More generally, are there transfers known on scissors congruence groups/pre-Bloch groups? As written above, these groups are defined in terms of points on $\mathbb{P}^1\setminus\{0,1,\infty\}$ modulo the five-term relation. A very naive approach to the definition of transfers for pre-Bloch groups in an extension $E/F$ would be to sum over $E$-points lying over $F$-points of $\mathbb{P}^1\setminus\{0,1,\infty\}$. Has anyone ever tried to work this out, or are there known obstruction why this cannot provide a transfer? Assuming it works, how would one relate such a naive definition to the definition of transfers for algebraic K-theory? In a related direction, what torsion elements besides those in $B(F)$ are known in the scissors congruence groups over fields which are not algebraically closed (in the algebraically closed case, the scissors congruence groups are uniquely divisible)?