[For part I see the author in ibid. 248, No. 1, 203-215 (2000; Zbl 0960.30015).] Let \(w= f(z)\) be a nonconstant meromorphic function on the complex plane \(\mathbb{C}\). It is well known that \(f\) is a Möbius transformation if and only if \(f\) maps circles in the \(z\)-plane onto circles in the \(w\)-plane, including straight lines among circles. It is also well known that: \(f\) is a Möbius transformation iff \(S_f(z)= 0\) for all \(x\in\mathbb{C}\setminus \{z: f'(z)= 0\}\) where \(S_f(z)\) is the Schwarzian derivative of \(f\). In this paper, the author gives some invariant characteristic properties of a certain class of Möbius transformations by means of their properties.