  Let $G$ be a Lie group equipped with a left invariant Randers metric of
Berward type $F$, with underlying left invariant Riemannian metric $g$. Suppose
that $\widetilde{F}$ and $\widetilde{g}$ are lifted Randers and Riemannian
metrics arising from $F$ and $g$ on the tangent Lie group $TG$ by vertical and
complete lifts. In this article we study the relations between the flag
curvature of the Randers manifold $(TG,\widetilde{F})$ and the sectional
curvature of the Riemannian manifold $(G,g)$ when $\widetilde{F}$ is of Berwald
type. Then we give all simply connected $3$-dimentional Lie groups such that
their tangent bundles admit Randers metrics of Berwarld type and their
geodesics vectors.
