  We exhibit an extension of the category of class two nilpotent groups. It has
the same objects but, unlike the latter, its morphisms are closed under
pointwise addition of maps. At the same time the class of its morphisms is much
smaller than the class of all maps. In fact, the morphisms are quadratic maps
of very special kind. It is indicated that classification of various class two
nilpotent groups up to isomorphism in this larger category can give interesting
results.
