homotopy theory - How do you compute the space of lifts of an E-infinity map?

The issue with the underlying monoid seems to complicate everything, in a similar way to how Postnikov decompositions are complicated by the $pi_1$ issue. In that case, the common technique is to fix $G = pi_1$ and consider the Postnikov tower as operating in the category of spaces over $K(G,1)$ rather than in the ordinary category of spaces.

So I don't see a lot of hope immediately for getting the $pi_0$ problem out of the way at the same time; it seems like it colors the whole problem.

Once you've decided on a lifting $pi_0 B to pi_0 X$, though, you can fix the underlying monoid because then you're reduced to studying lifts $B times_{pi_0 Y} pi_0 X to X$ over $pi_0 X$.

If you then fix $M = pi_0 X$ then there's certainly some kind of obstruction theory, but the problem is identifying the obstruction classes as coming from something cohomological that you can actually calculate. It seems to me that one should study the symmetric monoidal category of "spaces over $M$", with product having fibers
$$
(X star Y)_m = coprod_{m' m'' = m} (X_{m'} times Y_{m''})
$$
(which is some kind of left Kan extension), and try to get some handle on it.

Even when $M = mathbb{N}$ the bookkeeping gets complicated. Then you're studying "graded $E_infty$ spaces" and your obstruction theory will land in something like cohomology with coefficients in the relative homotopy groups of $Y$ over $B$, but you're taking cohomology of the "derived indecomposables" in your $E_infty$ space. The zero'th space of derived indecomposables of an $E_infty$ space $B$ over $mathbb{N}$ is the topological Andr'e-Quillen homology object of $B_0$. Even if $B_0$ is trivial, then the zero'th derived indecomposable space is trivial, the first is $B_1$, and the next is the homotopy cofiber of the squaring map $(B_1 times B_1)_{hmathbb{Z}/2} to B_2$.

Based on this kind of futzing around I am led to believe that your obstructions may possibly occur in the relative topological Andre-Quillen cohomology of $Sigma^infty_+ B$ over $Sigma^infty_+ M$ with coefficients in the relative homotopy of $X$ over $Y$. But the problem seems very difficult for a general monoid $M$.

(Especially evidenced by the fact that Charles Rezk hasn't popped in here with an answer yet.)