(1) Suppose $\pi$ is a set of primes and $G$ is a $\pi$-divisible nilpotent group, i.e., for any $g \in G$ and $p \in \pi$, there exists $x \in G$ such that $x^p = g$. Is it necessary that all the homology groups of $G$ for the trivial group action with coefficients in $\mathbb{Z}$ are also $\pi$-divisible groups? I am in particular interested in the second homology group $H\_2(G;\mathbb{Z})$, which is the Schur multiplier of $G$. Note that the result is true for $H\_1(G;\mathbb{Z})$, which is the abelianization of $G$.


I believe that the following example shows that the result is false for $H\_2(G;\mathbb{Z})$, but I'm not sure whether I am applying results correctly. The example group $G$ that I have in mind is defined as follows. Let $UT(3,\mathbb{Q})$ denote the group of $3 \times 3$ upper triangular matrices over the rational numbers that have 1s on the diagonal. Let $Z$ be an infinite cyclic subgroup of $UT(3,\mathbb{Q})$ generated by any non-identity element in its center. Let $G = UT(3,\mathbb{Q})/Z$. Then, $G$ is divisible by all primes, on account of being a quotient of $UT(3,\mathbb{Q})$, which is divisible by all primes. $G$ admits $UT(3,\mathbb{Q})$ as a stem extension group and the base of the extension is $Z$, isomorphic to $\mathbb{Z}$, which is not divisible by any prime. The base of any stem extension is a quotient of the Schur multiplier, so the Schur multiplier (which I haven't computed precisely) should not be divisible by any prime.


(2) Suppose $\pi$ is a set of primes and $G$ is a $\pi$-powered nilpotent group, i.e., for any $g \in G$ and $p \in \pi$, there exists unique $x \in G$ such that $x^p = g$. Is it necessary that all the homology groups of $G$ for the trivial group action with coefficients in $\mathbb{Z}$ are also $\pi$-powered groups? I am in particular interested in the second homology group $H\_2(G;\mathbb{Z})$, which is the Schur multiplier of $G$. Note that the result is true for $H\_1(G;\mathbb{Z})$, which is the abelianization of $G$ (the proof of this uses that $G$ is nilpotent).


Can anything be said in general, in the non-nilpotent case?



Homology groups of divisible and powered (nilpotent) groups
In the nilpotent case - YES:
Let $\pi$ be any set of primes. A group $K$ is called $\pi$-local if for every $n$ with no prime divisors in $\pi$ the $n$th power function $k\mapsto k^n$ is a bijection $K\to K$. If $G$ is any nilpotent group then there exists a $\pi$-localization of $G$ - it is the initial homomorphism 
$$h:G\to G\_\pi$$ 
into a $\pi$-local group. Initial means that if $k:G\to K$ is any homomorphism with $K$ being $\pi$-local then $k$ uniquely factors through $h$. Now every $\pi$ -localization of a nilpotent group induces a $\pi$-localization on the reduced homology
 $$h\_\ast:H\_\ast(G;\mathbb{Z})\to H\_\ast(G\_\pi;\mathbb{Z})$$
Your question follows.
You can find the details in
P. Hilton, G. Mislin and J. Roitberg. Localization of Nilpotent Groups and Spaces. North-Holland Math. Studies no. 15 (North-Holland, 1975).


I am not aware of any so elegant results along these lines for more general groups, you may read about virtually nilpotent groups in
C. Casacuberta and M. Castellet. Localization methods in the study of the homology of virtually nilpotent groups. Math. Proc. Cambridge Philos. Soc. 112 (1992), no. 3, 551–564.


