The lifting problem in group cohomology (Q1919566)

Let \(\mathbf G\) be a profinite group, let \(M\) be a discrete continuous \(\mathbf G\)-module and let \(q>1\) be an integer. Assume that \(H^q({\mathbf G},M)=0\). Then for a given finite quotient group \(G={\mathbf G}/{\mathbf H}\) of \(\mathbf G\) there is a finite group extension \(G^*\) of \(G\) such that the corresponding embedding problem for \(\mathbf G\) is solvable, say \(G^*={\mathbf G}/{\mathbf J}\), and such that the inflation homomorphism \(\inf:H^q(G,M^{\mathbf H})\to H^q(G^*,M^{\mathbf J})\) is trivial. In such a situation \(G^*\) is called a lifting group for \(H^q(G,M^{\mathbf H})\). The purpose of this note is to obtain some results about lifting groups for a profinite group \(\mathbf G\) such that the cohomological dimension of \(\mathbf G\) is not greater than 1 or such that the strict cohomological dimension of \(\mathbf G\) is not greater than 2. As will be pointed out, the case of cohomological dimension not greater than 1 yields a generalization of I. Schur's concept of representation group and the case of strict cohomological dimension not greater than 2 has some relations to algebraic number theory.