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Some questions about splitting of sequence $0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$ for Henselian val field $K$

(1) Since $k/k'$ is totally inseparable, the natural map $\mathrm{Gal}(k') \to \mathrm{Gal}(k)$ is an isomorphism, so if we have a splitting $\mathrm{Gal}(k') \to \mathrm{Gal}(K')$, the composition $$...
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