  We study an impartial game introduced by Anderson and Harary. The game is
played by two players who alternately choose previously-unselected elements of
a finite group. The first player who builds a generating set from the
jointly-selected elements wins. We determine the nim-numbers of this game for
finite groups of the form $T \times H$, where $T$ is a $2$-group and $H$ is a
group of odd order. This includes all nilpotent and hence abelian groups.
