  We study the $\mathbb{Z}_2$-homology groups of the orbit space $X_n =
G_{n,2}/T^n$ for the canonical action of the compact torus $T^n$ on a complex
Grassmann manifold $G_{n,2}$. Our starting point is the model $(U_n, p_n)$ for
$X_n$ constructed by Buchstaber and Terzi\'c (2020), where $U_n = \Delta
_{n,2}\times \mathcal{F}_{n}$ for a hypersimplex $\Delta_{n,2}$ and an
universal space of parameters $\mathcal{F}_{n}$ defined in Buchstaber and
Terzi\'c (2019), (2020). It is proved by Buchstaber and Terzi\'c (2021) that
$\mathcal{F}_{n}$ is diffeomorphic to the moduli space $\mathcal{M}_{0,n}$ of
stable $n$-pointed genus zero curves. We exploit the results from Keel (1992)
and Ceyhan (2009) on homology groups of $\mathcal{M}_{0,n}$ and express them in
terms of the stratification of $\mathcal{F}_{n}$ which are incorporated in the
model $(U_n, p_n)$. In the result we provide the description of cycles in
$X_n$, inductively on $ n. $ We obtain as well explicit formulas for
$\mathbb{Z}_2$-homology groups for $X_5$ and $X_6$. The results for $X_5$
recover by different method the results from Buchstaber and Terzi\'c (2021) and
S\"uss (2020). The results for $X_6$ we consider to be new.
