Free action on integral homology spheres Motivation: The cyclic group with two elements can freely act on $\mathbb{S}^1$. Note that $\mathbb{S}^1$ is also the Eilenberg–MacLane space $K(\mathbb{Z}, 1)$.
Question Let $n\geq 3$. Is there any integral-homology $n$-sphere such that it admits a free $\mathbb{Z}_2$-action and it is also an Eilenberg–MacLane space $K(G,1)$ for some group $G$?
 A: I will assume that by a (integral) homology $n$-sphere you mean an $n$-dimensional manifold $M$ which has $H_*(M)\cong H_*(S^n)$.  (One can consider instead of manifolds more general CW complexes.) According to Thurston's version of the Hopf Conjecture, if $M$ is a closed aspherical manifold of dimension $2k$ then
$$
(-1)^k \chi(M)  \ge 0.
$$
In particular, if $k$ is odd, then, conjecturally, $\chi(M)\le 0$. Such a manifold cannot be a homology sphere.
On the other hand, there is an abundant supply of closed aspherical 3-manifolds which are integral 3-spheres. Early examples of these are given by Seifert manifolds whose base-orbifold has nonpositive Euler characteristic. For every Seifert manifold $M$ whose Seifert fibration is orientable, we have a canonical $S^1$-action on $M$, corresponding to the rotation of the fibers. This action is not free if there are singular fibers. However, suppose that Seifert invariants $(\alpha_i,\beta_i)$ of singular fibers are all such that $\alpha_i$ is coprime with  $n$, then the order $n$ rotation of the fiber acts freely on $M$. It remains to find examples $M$ which are integral homology spheres. These are given, for instance, by Brieskorn manifolds $M(p,q,r)$ for which $p, q, r$ are all coprime. A very nice exposition of Brieskorn 3-manifolds is given in Milnor's paper
Milnor, John W., On the 3-dimensional Brieskorn manifolds M(p,q,r), Knots, Groups, 3-Manif.; Pap. dedic. Mem. R. H. Fox, 175-225 (1975). ZBL0305.57003.
Taking $p, q, r$ which are all odd and coprime (say, $3, 5, 7$) yields an example of a 3-dimensional integral homology sphere which is aspherical and admits a free involution.
I am sure there are other examples in all odd dimensions, but sorting this out would take some work.
I am not sure what happens if you allow arbitrary CW complexes as your $K(G,1)$'s, I suspect that you get examples in even dimensions as well.
There is a separate interesting question regarding which finite groups admit free actions on integral homology spheres. For some results in this direction, see
Milnor, John W., Groups which act on (S^ n) without fixed points, Am. J. Math. 79, 623-630 (1957). ZBL0078.16304.
and
Wall, C. T. C., Free actions of finite groups on spheres, Algebr. geom. Topol., Stanford/Calif. 1976, Proc. Symp. Pure Math., Vol. 32, Part 1, 115-124 (1978). ZBL0407.57031.
