Fixed points of diffeomorphisms on nilmanifolds with a free nilpotent fundamental group

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.