Assume that $G,H$ are two sheaves of groups (say in fpqc topology on the scheme $X$) and there is a map $G\to H$ which is representable by a closed immersion. Let us also assume that the quotient is representable by a scheme $H/G$. under what condition we can prove the representability of the map between the stacks $Bun_{G,X}\to Bun_{H,X}$?
I think in general this should be related to the representability of zero locus of a class in $H^1(X,H/G)$, but I'm not sure about the details. Can anyone explain this or point me to the literature on the subject?
