  We pursuit the research line proposed in \cite{YZ-Gflat} about the
classification of Hermitian manifolds whose $s$-Gauduchon connection $\nabla^s
=(1-\frac{s}{2})\nabla^c + \frac{s}{2}\nabla^b$ is flat, where $s \in
\mathbb{R}$ and $\nabla^c$ and $\nabla^b$ are the Chern and the Bismut
connections, respectively. We focus on Lie groups equipped with a left
invariant Hermitian structure. Such spaces provide an important class of
Hermitian manifolds in various contexts and are often a valuable vehicle for
testing new phenomena in complex and Hermitian geometry. More precisely, we
consider a connected $2n$-dimensional Lie group $G$ equipped with a
left-invariant complex structure $J$ and a left-invariant compatible metric $g$
and we assume that its connection $\nabla^s$ is flat. Our main result states
that if either $n$=2 or there exits a $\nabla^s$-parallel left invariant frame
on $G$, then $g$ must be K\"ahler. This result demonstrates rigidity properties
of some complete Hermitian manifolds with $\nabla^s$-flat Hermitian metrics.
