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By the inflation-restriction exact sequence this will be true precisely when $H^1(L, G)^{\text{Gal}\left(L/F\right)}$ is trivial. The superscript "Gal(L/F)" there means to take invariants under an action you define by hand on $H^1(L, G)$. [EDIT: An old version of this post said "precisely when," which is not correct. Thanks!]

This may not be helpful, since it sounds like your point is that you want to avoid working with direct limits. But I'd do that just by requiring that cochains be continuous.
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By the inflation-restriction exact sequence this will be true precisely when H^1(L, G)^Gal(L/F) $H^1(L, G)^{\text{Gal}\left(L/F\right)}$ is trivial. The superscript "^Gal(L/F)" Gal(L/F)" there means to take invariants under an action you define by hand on H^1(L, G)$H^1(L, G)$.

This may not be helpful, since it sounds like your point is that you want to avoid working with direct limits. But I'd do that just by requiring that cochains be continuous.
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By the inflation-restriction exact sequence this will be true precisely when H^1(L, G)^Gal(L/F) is trivial. The superscript "^Gal(L/F)" there means to take invariants under an action you define by hand on H^1(L, G).

This may not be helpful, since it sounds like your point is that you want to avoid working with direct limits. But I'd do that just by requiring that cochains be continuous.