Let $K$ be a henselian valuation field with residue field $k$, then the decomposition group surjects onto Galois group of the residue field, with kernel the inertia subgroup, namely we have short exact sequence:$$0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$$

When $K$ is a local field, we can split the sequence by lifting the Frobenius; when $K=k((t))$, we can split the sequence by lifting the Galois action (with trivial action on $t$). But in general, do we know if the sequence always split?

(The splitting of the sequence is mentioned as a well-known fact in Proposition A5 in "Exposant et indice d'algèbres simples centrales non ramifiées", but I couldn't find a reference.. Splitting of the sequence would imply that the restriction morphism $H^i_{et}(\mathrm{Spec}(V),M)\to H^i_{et}(\mathrm{Spec}(K),M)$ is injective for any locally constant sheave $M$ on the valuation ring $V$.)