We pursuit the research line proposed in [24] about the classification of Hermitian manifolds whose s-Gauduchon connection ∇ s = (1 − s 2 )∇ c + s 2 ∇ b is flat, where s ∈ Ê and ∇ c and ∇ 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 leftinvariant compatible metric g and we assume that its connection ∇ s is flat. Our main result states that if either n=2 or there exits a ∇ s -parallel left invariant frame on G, then g must be Kähler. This result demonstrates rigidity properties of some complete Hermitian manifolds with ∇ s -flat Hermitian metrics.In the following given a real number s, we call ∇ s as the s-Gauduchon connection. Note that ∇ 0 = ∇ c and ∇ 2 = ∇ b . Moreover, ∇ 1 corresponds to the projection of the Levi-Civita connection onto the space of (1, 0)-vector fields. This connection is called the first canonical Hermitian connection in [8] and [13], the associated connection in [9] and the complexified Levi-Civita connection in [12]. In [24], it is proposed the study of compact Hermitian manifolds with flat s-Gauduchon connections, as a special case of a problem posed by Yau (Problem 87 in [26]). The case s = 0 (i.e. when the Chern connection is flat) was studied in the 1950s by W. Boothby [5] and H. C. Wang [20]. Boothby proved that if (M n , g) is a compact Chern-flat 2010 Mathematics Subject Classification. 53C55 (Primary).