Let $p$ be a prime number, $\mathbb{F}_p$ the field with $p$ element and $\omega$ the mod $p$ cyclotomic character. Let $K$ be a finite extension of $\mathbb{Q}_p$ (the field of $p$-adic numbers) and denote by $G_K$ the absolute Galois group of $K$. Let $V$ be a $2$ dimensional $\mathbb{F}_p$-representation of $G_K$ which is a non-split extension of $\omega$ by $\omega$. So we have a short exact sequence $1 \to \omega \to V \to \omega \to 1$ of $\mathbb{F}_p$-representations inducing a map in Galois cohomology : $H^1(G_K, V) \to H^1(G_K, \omega)$.
Is this map surjective ? If not, can we describe its image (in terms of peu ramifiées or très ramifiées extensions) ?
EDIT : Well it turns out this map is not surjective : the map $H^2(G_K, V) \to H^2(G_K, \omega)$ is an isomorphism (it is surjective because $H^3 = 0$ and those are 1 dimensional vector spaces (it is easily seen using Tate local duality)). It implies that the map $H^1(G_K,\omega) \to H^2(G_K, \omega)$ is surjective with non trivial kernel. So I guess the only question it remains to answer is if it is possible to describe this non trivial kernel.