There are lots of natural functors (that define sheaves in the fppf topology) that are not representable by schemes. For example, hilbert schemes of proper non-projective schemes in general need algebraic spaces. However, I know of no examples of such subtleties with group schemes. Every sheaf of groups that I know of is already representable.

Is this a consequence of general theorems? Is it considered easier to show that a sheaf of groups (in the fppf topology) is representable by a scheme than a sheaf of sets? For example, are necessary and sufficient criteria known for a sheaf of groups in the fppf topology to be representable. How about for sheaves of abelian groups?