Let $S$ be a scheme, $(\text{Sch}/S)_{\rm Ét}$ a big étale site, and $A$ a representable (either in schemes or algebraic spaces over $S$) abelian sheaf on $(\text{Sch}/S)_{\rm Ét}$.
Suppose there is a short exact sequence of abelian sheaves on $(\text{Sch}/S)_{Ét}$:
$$0\to A_1\to A\to A_2\to 0.$$
Are $A_1$ and $A_2$ representable?
are the maps $A_1\to A$ and $A\to A_1$ representable, the former by a closed immersion?
If both maps are representable, then $A_1$ is representable if and only if $A_2$ is.
Conversely:
if $A_1$ and $A_2$ are representable, is $A$ too?
In other words, is the full subcategory of the category of abelian sheaves on $(\text{Sch}/S)_{Ét}$ given by representable abelian sheaves and morphisms/representable morphisms of abelian sheaves, a weak Serre subcategory?
