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One approach: bash it out: take the "universal" representation of $S_4$

It seems according to a local $k$ algebra where $2 = 0$, and show Tim that the image is Artinian.


An unreliable computation: by inflation-restriction, $$0 \rightarrow H^1(\Z/2\Z,W^{\Z/3\Z}) \rightarrow H^1(S_3,W) \rightarrow H^1(\Z/3\Z,W).$$ The last term is zero, and $W^{\Z/3\Z}$ is the regular representation for $S_3$, so $H^1(S_3,W) = 0$. Then, by inflation restriction, $$H^1(S_4,W) \hookrightarrow H^1(K,W)^{S_3}.$$

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