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

Choose a normalized set-theoretic section $s:H\to E$ of $\pi$, so $s(1)=1$. Define the extension cocycleAssociativity of $s(h)s(k)s(l)$ gives, in multiplicative notation for $M$,which is exactly the two-cocycle identity. Thus $\varphi$ represents the class of the group extension in $H^2(H,M)$.