Non-splitting of the Tor extension in the Bloch–Wigner-type sequences for PGL2

Establish that, for any domain A, the short exact sequence 0 → Tor_1^Z(µ(A), µ(A)) → E → Z/2 → 0 appearing as the Tor term in the Bloch–Wigner-type exact sequences for H_3(PGL_2(A), Z) in Propositions 10.9 and 10.10 is non-split; equivalently, prove that the associated extension class in Ext^1_Z(Z/2, Tor_1^Z(µ(A), µ(A))) is nontrivial, where µ(A) denotes the group of roots of unity in A.

Background

In Section 10 the authors derive Bloch–Wigner-type exact sequences for H_3(PGL_2(A), Z) over several classes of rings (non-dyadic local fields and certain rings such as Z and Z[1/2]). In these sequences, a Tor-term occurs that is described as an extension of Z/2 by Tor_1^Z(µ(A), µ(A)).

Remark 10.11 discusses the classification of such extensions via Ext^1_Z(Z/2, Tor_1^Z(µ(A), µ(A))) and notes that, up to isomorphism, there are at most two possibilities (split or non-split). The authors state their belief that the specific extension arising in Propositions 10.9 and 10.10 is the non-split one but explicitly acknowledge they do not have a proof.