Consider the following question about forms of a given group that are embedded in a fixed group.

Fix for simplicity $k$ a perfect field, and $H\subset G$ a pair of connected reductive $k$-groups, with $G$ being $k$-simple. It is possible that one finds other $k$-groups $L\subset G$ that are $k$-forms of $H$.

(1) How to classify those $k$-forms $L$ embedded in $G$ such that $L_\bar{k}$ is conjugate to $H_{\bar{k}}$ by $G(\bar{k})$?

(2) Given a $k$-form $L$ of $H$, can one always find a larger $k$-group $G$ containing both $H$ and $H'$, such that (1) holds for the triple $(H,L,G)$? Here $G$ is required to not normalize any $k$-factor of $H$.

Thanks a lot.