preserving homology

At my ftp site ftp.math.mcgill.ca/pub/barr/pdffiles is a note called preservinghomology.pdf that raises (but does not satisfactorily answer) the question of what it means for an additive functor between abelian categories to preserve homology. It does work for functors that are either right or left exact. If anyone has solved (or even looked seriously at the question) I would appreciate hearing from them. Michael