I am interested in answers or reference in the literature to the following problem:

Classify up to homotopy all maps $A\to X$, where $A$ is a closed oriented manifold and $X$ is a closed nilmanifold of the same dimension as $A$.

Of course by "classify'' here we mean describe a complete set of invariants to distinguish between maps up to homotopy, for instance in terms of cohomology.

Recall that a nilmanifold is an aspherical manifold whose fundamental group is nilpotent and torsion-free. One can build its refined Postnikov tower directly from the lower central series of the fundamental group, as in this question. This exhibits the nilmanifold as an iterated principal torus bundle.

For this reason I feel that the problem may be approachable. For instance, if $X$ is an abelian nilmanifold (ie a torus) then a map $A\to X$ is classified by a tuple of elements of $H^1(A;\mathbb{Z})$.

Meanwhile, I am well aware that solutions to problems of this type usually exist only in the case where $X$ is simply-connected, and without this assumption things become considerably more complicated.

I would be happy with an answer in the simplest non-trivial case, ie when $X$ is the $3$-dimensional Heisenberg manifold.