Associated torsor via inclusion of sheafs

Question

Suppose we have a scheme $S$ and two sheaves of groups $H, H'$ on $S$ for the fpqc topology such that $H \subset H'$. Let $\mathcal{H}$ be an $H$-torsor. Then how do you naturally get an $H'$-torsor associated to $\mathcal{H}$?

Answer 1

You take the sheafification of

$$S\mapsto (H'(S)\times\mathcal{H}(S))/\sim$$

where $(h'_1,h_1)\sim (h'_2,h_2)$ if there exists some $h\in H(S)$ such that $(h_2',h_2)=(h_1' h,h^{-1}h_1)$.