  We consider $A$-hypergeometric (or GKZ-)systems in the case where the grading
(character) group is an arbitrary finitely generated Abelian group. Emulating
the approach taken for classical GKZ-systems in arXiv:math/0406383 that allows
for a coefficient module, we show that these $D$-modules are holonomic systems.
For this purpose we formulate an Euler--Koszul complex in this context, built
on an extension of the category of $A$-toric modules. We derive that these new
systems are regular holonomic under circumstances that are similar to those
that lead to regular holonomic classical GKZ-systems.
  For the appropriate coefficient module, our $D$-modules specialize to the
"better behaved GKZ-systems" introduced by Borisov and Horja. We certify the
corresponding $D$-modules as regular holonomic, and establish a holonomic
duality on the level of $D$-modules that was suggested on the level of
solutions by Borisov and Horja and later shown by Borisov and Han in a special
situation (arXiv:1308.2238, arXiv:2301.01374).
