Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 5 / a / ii / Solution

The fiber product of groupsis a group under componentwise multiplication. The maps $m\mapsto (m,1)$ and $(e,g)\mapsto g$ give an exact sequenceThe section $g\mapsto (s(f(g)),g)$ has extension cocycleTherefore this pullback extension represents $f^{\ast }([\varphi ])\in H^2(G,M)$.