Self-dual, self-Petrie-dual and M\"obius regular maps on linear fractional groups

Regular maps on linear fractional groups $PSL(2,q)$ and $PGL(2,q$) have been studied for many years and the theory is well-developed, including generating sets for the asscoiated groups. This paper studies the properties of self-duality, self-Petrie-duality and M\"obius regularity in this context, providing necessary and sufficient conditions for each case. We also address the special case for regular maps of type (5,5). The final section includes an enumeration of the $PSL(2,q)$ maps for $q\le81$ and a list of all the $PSL(2,q)$ maps which have any of these special properties for $q\le49$.