Spaces that are finitely covered by manifolds

Indeed, C. Thomas, 3-Manifolds and PD(3)-groups, in "Novikov conjectures", Vol. 2, gives a reference to Swan's example. Namely, there is a finite complex $X$ which is a homotopy 3-sphere, a free $S_3$-action $S_3\times X\to X$, which cannot be realized (up to homotopy) as a free $S_3$-action on $S^3$, since $S_3$ cannot act freely on $S^3$ (the 3-sphere).

Conditions where $N$ exists are for example when $M$ is a non-positively curved locally symmetric manifold, by a theorem of Mostow (for hyperbolic manifolds) and Margulis in general. Mostow's theorem is generalized to manifolds with hyperbolic fundamental group whose Gromov boundary is a sphere of dimension at least 5. So if one started with such a manifold, any group containing the fundamental group with finite index and torsion-free would have to be a manifold group.

More generally, it is conjectured that finitely presented groups which are $PD(n)$ groups are the fundamental groups of closed $n$-manifolds (Conjecture 3.4). If true, this would show that $N$ exists when $X$ is aspherical. This conjecture would follow for $n\geq 5$ from the Novikov conjecture. Many cases of this conjecture are known (for groups satisfying certain restrictions). See also Sections 5 and 8 of this paper.