This question is a followup to Serre's Exercise 3.4.1 on Galois Cohomology. Let $G$ be a profinite group, $1 \to P \to E \to W \to 1$ a finite group extension and $f\colon G \to W$ a continuous surjection. I want to show the equivalence of the two following properties:

  1. Whenever $P$ is an abelian group killed by $p$ and $E \simeq P \rtimes W$, there is a surjective lifting $f'\colon G \to E$ of $f$.
  2. For every open normal subgroup $U$ of $G$, and for any integer $N \geq 0$, there exist $z_1,\cdots,z_N \in H^1(U, \mathbb{Z}/p\mathbb{Z})$ such that the elements $s(z_i)$ ($s \in G/U$, $1 \leq i \leq N$) are linearly independent over $\mathbb{Z}/p\mathbb{Z}$.

THE ATTEMPTS SO FAR I would guess this has to do with the five-term exact sequence. Also, I'm not sure if the pullback $1 \to P \to E_f \to G \to 1$ of the split extension $1 \to P \to E \to W \to 1$ also splits (though I would guess so). If this is the case, maybe I can get something out of $H^2(G, P)$.


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