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Regular maps on linear fractional groups $PSL(2,q)$ and $PGL(2,q$) have beenstudied for many years and the theory is well-developed, including generatingsets for the asscoiated groups. This paper studies the properties ofself-duality, self-Petrie-duality and M\"obius regularity in this context,providing necessary and sufficient conditions for each case. We also addressthe special case for regular maps of type (5,5). The final section includes anenumeration 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 $q\le49$.

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