Does $PGL(2,\Bbb{C})$ compose all circle-preserving topological automorphisms of Riemann sphere $\overline{\Bbb{C}}$? This question arises from the fact that $PGL(3,\Bbb{R})$ composes all line-preserving topological automorphisms of $\Bbb{R}\Bbb{P}^2$, but the latter (according to my poor knowledge) is not easy to prove.
Edit: I forgot to restrict the automorphisms to be orientation-preserving.
Theorem. Suppose that $f: S^2\to S^2$ is a bijection which maps circles to circles. Then $f$ is a Moebius transformation, i.e. either is a linear-fractional transformation (an element of $PSL(2,{\mathbb C})$) or a composition of a linear-fractional transformation with the complex conjugation (or inversion). See section 6.7 in
Coxeter, H. S. M., Introduction to geometry. 2nd ed, New York etc.: John Wiley and Sons, Inc. XVI, 469 p. (1969). ZBL0181.48101.
The same result holds for bijections of higher-dimensional spheres (which send codimension 1 subspheres to codimension 1 subspheres): These are all Moebius transformations, i.e. compositions of inversions. See the same book, section 7.7.
Lastly, regarding the fact that line-preserving bijective self-maps of $RP^2$ are projective transformations, the proof is clever but not at all difficult. One can find a proof for instance in Hartshorne's book on projective geometry.