How to construct a CW complex $X$ with prescribed homotopy groups $\pi_i(X)$ and prescribed actions of $\pi_1(X)$ on the $\pi_i(X)$'s?
One method is as follows.
Of course, you don't have to take $Y$ to be a product of Eilenberg-Mac Lane spaces. Passing between universal covers and Borel constructions reduced your original question to finding a simply-connected space with the given higher homotopy groups and an action of $\pi_1$ on the whole space.
You might also specify more structure; for example, the higher homotopy groups in the outlined construction are not related by Whitehead products or anything similar. This makes the problem harder. If you specify all the structure that the homotopy groups naturally have, you have a so-called $\Pi$-algebra; the paper "The realization space of a $\Pi$-algebra" by Blanc-Dwyer-Goerss studies this problem and how one might go about classifying realizations.