  We extend the methods of geometric invariant theory to actions of
non--reductive groups in the case of homomorphisms between decomposable sheaves
whose automorphism groups are non--reductive. Given a linearization of the
natural action of the group $\Aut(E)\times\Aut(F)$ on Hom(E,F), a homomorphism
is called stable if its orbit with respect to the unipotent radical is
contained in the stable locus with respect to the natural reductive subgroup of
the automorphism group. We encounter effective numerical conditions for a
linearization such that the corresponding open set of semi-stable homomorphisms
admits a good and projective quotient in the sense of geometric invariant
theory, and that this quotient is in addition a geometric quotient on the set
of stable homomorphisms.
