  Let $M$ be a nilmanifold with a fundamental group which is free $2$-step
nilpotent on at least 4 generators. We will show that for any nonnegative
integer $n$ there exists a self-diffeomorphism $h_n$ of $M$ such that $h_n$ has
exactly $n$ fixed points and any self-map $f$ of $M$ which is homotopic to
$h_n$ has at least $n$ fixed points. We will also shed some light on the
situation for less generators and also for higher nilpotency classes.
