  The contributions of $G'$-boson predicted by the chiral color symmetry of
quarks to the cross section $\sigma_{t\bar{t}}$ and to the forward-backward
asymmetry $A_{\rm FB}^{p \bar p}$ of $t\bar{t}$ production at the Tevatron are
calculated with account of the difference of the strengths of the $\bar q q G$
and $\bar q q G'$ interactions. The results are analysed in dependence on two
free parameters of the model, the mixing angle $\theta_G$ and $G'$ mass
$m_{G'}$. The $G'$-boson contributions to $\sigma_{t\bar{t}}$ and $A_{\rm
FB}^{p \bar p}$ are shown to be consistent with the Tevatron data on
$\sigma_{t\bar{t}}$ and $A_{\rm FB}^{p \bar p}$, the allowed region in the
$m_{G'} - \theta_G$ plane is discussed and around $m_{G'}=1.2 \, TeV, \;
\theta_G=14^\circ $ the region of $1 \sigma$ consistency is found.
