Let $K/k$ be an extension of fields, not necessarily algebraic; let $G$ and $H$ be split, reductive groups over $K$; and let $f : H \to G$ be an embedding of groups.

Do there exist split, reductive groups $G'$ and $H'$ over $k$, an embedding $f' : H' \to G'$ of groups, and isomorphisms $G'_K \cong G$ and $H'_K \cong H$ such that $f'_K$ is identified to $f$?

If it helps, $k$ and $K$ can both be assumed algebraically closed. (I would not be surprised if this assumption is necessary, but I would also not be surprised if just being split is enough.)

(I could ask this question with fixed $k$-groups $H'$ and $G'$ at the beginning, consider a morphism $f : H'_K \to G'_K$, and then ask for $f'$, but in that case the answer is ‘no’; for example, take $H' = \operatorname{GL}_1$ and $G' = \operatorname{SL}_2$, and let $f$ be any embedding of $H'_K$ as a maximal split torus in $G'_K$ that is not defined over $k$. If I did not require that $G$ and $H$ be split over $K$, then the answer would be ‘no’ just because one or both of them might not admit a $k$-form.)

This seems like it's in the spirit of Borel and Tits - Homomorphismes “abstraits” …, but I couldn't find it there or deduce it from the results of that paper.

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