Fix a field $K$ and a separable closure $K^{sep}$ of $K$. Consider the category of all separable field extensions $K \subset L \subset K^{sep}$ with field inclusions as morphisms. Consider the first Galois cohomology functor from this category to the category of Sets, which sends $L \mapsto H^1(Gal(K^{sep}/L), G(K^{sep}))$ for some fixed $G(K^{sep})$ and where any morphism $f: L \to F$ is sent to the induced morphism $f: H^1(Gal(K^{sep}/L), G(K^{sep})) \to H^1(Gal(K^{sep}/F), G(K^{sep}))$. My question is:
Given a pullback diagram in the category defined above $$ \begin{array}{ccc} E \times_L F &\to &F\\ \downarrow & & \downarrow\\ E & \to & L \end{array} $$ under which conditions does the first Galois cohomology functor preserve the pullback diagram?
Additional properties that I am happy to assume if it makes things easier:
- In the pullback diagram, the inclusions $F \subset L$ and $E \subset L$ are finite extensions.
- The fields involved are number fields (i.e. finite extensions of $\mathbb{Q}$).