Let $i : Z \to X$ be a closed immersion of schemes. Is $i_* : Ab((Sch/Z)_{fppf}) \to Ab((Sch/X)_{fppf})$ an exact functor?

The answer is yes in the \'etale or syntomic topology. It seems likely the answer is no in the fppf case however.

In terms of algebra, the problem is related to the following question. Suppose that $A$ is a ring and $I$ is an ideal. Let $A/I \to C$ be a faithfully flat, finitely presented ring map. Is there a flat, finitely presented ring map $A \to B$ with $B/IB$ nonzero such that $A/I \to B/IB$ factors through $C$?

In fact, if one finds an example $(A, I, C)$ for which there does **not** exist any $B$, then one has a counter example to exactness of $i_*$.