  Let $H_n$ be the $n$-th group homology functor (with integer coeffcients) and
let $\{G_i\} _ {i \in \mathbb{N}}$ be any tower of groups such that all maps
$G_{i+1} \to G_i$ are surjective. In this work we study kernel and cokernel of
the following natural map: $$H_n(\varprojlim G_i) \to \varprojlim H_n(G_i)$$
For $n=1$ Barnea and Shelah [BS] proved that this map is surjective and its
kernel is a cotorsion group for any such tower $\{G_i\} _ {i \in \mathbb{N}}$.
We show that for $n=2$ the kernel can be non-cotorsion group even in the case
when all $G_i$ are abelian and after it we study these kernels and cokernels
for towers of abelian groups in more detail.
