I am wondering whether the analogue of III.1.27 in Milne's "Etale cohomology" holds true if one works with fppf cohomology with supports instead of etale cohomology with supports. More precisely, let $f\colon X^{\prime} \rightarrow X$ be fppf, let $Z \hookrightarrow X$ be a closed subscheme, and let $Z^{\prime} \hookrightarrow X^{\prime}$ be its preimage, i.e., $Z^{\prime} = Z \times_X X^{\prime}$. Suppose that $f|_{Z^{\prime}}\colon Z^{\prime} \rightarrow Z$ is an isomorphism and also that $f(X^{\prime} - Z^{\prime}) \subset X - Z$. Let $F$ be a sheaf on $X_{fppf}$. Is it true that the natural morphism of fppf cohomology groups with supports $H^i_{Z}(X, F) \rightarrow H^i_{Z^{\prime}}(X^{\prime}, f^*F)$ is an isomorphism?
As in loc. cit., one is reduced to considering $i = 0$, in which case the proof of injectivity continues to work. Imitating the proof of surjectivity in loc. cit., one would take $\gamma^{\prime} \in \Gamma_{Z^{\prime}}(X^{\prime}, F) \subset \Gamma(X^{\prime}, F)$ and would attempt to glue it with the zero section on $X - Z$. To check the glueing condition, one would need to check in particular that the two pull-backs of $\gamma^{\prime}$ to $X^{\prime} \times_X X^{\prime}$ agree. This step is not explained explicitly in loc. cit., presumably because it is clear in the view of II.3.11. The latter says that the push-forward establishes an equivalence of the category of etale sheaves on $Z$ onto the full subcategory of etale sheaves on $X$ consisting of those sheaves that are supported on $Z$. That brings me to my second question: is there an analogue of this result for the fppf topology?
