Word images and their impostors in finite nilpotent groups

In 2014, Lubotzky showed that in a finite simple group G every automorphism invariant subset containing the identity is the image of a word map on G. We show that the number of such subsets in a finite nilpotent group may be arbitrarily larger than the number of word images. We construct a 2-exhaustive set of word maps on nilpotent groups of class 2 and demonstrate its minimality in some cases. We also find a sufficient condition, in terms of restrictions on word images, for a group to be a special p-group.