This is probably a stupid question, but I can't figure it out.
Let $G, G', G''$ be linear algebraic groups over a characteristic $0$ field $K$. Say that $G' \in A$ and $G'' \in B$, where $A,B$ are subsets of the set of linear algebraic $K$-groups such that $A,B$ are closed under taking $K$-forms. Suppose also that there is a short exact sequence $1 \to G' \to G \to G'' \to 1$. Now let $H$ be a $K$-form of $G$.
Is it true that there is a short exact sequence $1 \to H' \to H \to H \to 1$ such that $H' \in A$ and $H'' \in B$?