"Let $G$ be a transitive group of permutations on a given set of letters. Let a new fixed letter be adjoined to every permutation of $G$. Then a transitive group $H$ of permutations on the combined set of letters is called a *transitive extension* of $G$ if it contains $G$ as the largest subgroup fixing the new letter."


... T. C. Holyoke, *Transitive extensions of dihedral groups*, Math. Zeitschr. Bd. 60, S. 79/80 (1954)


In this paper, Holyoke determined that the transitive extensions of a dihedral group $D\_n$ acting in the usual way on $n$ points, are limited to the extensions: $S\_3$ (of $D\_2$), $S\_4$ (of $D\_3$) and $PSL(2, 5)$ of $D\_5$. My question is, what are all the transitive extensions of cyclic groups?


EDIT: Here is a modern equivalent formulation of the question: "What are the $2$-transitive groups with a cyclic point-stabiliser?"



What are all the transitive extensions of cyclic groups?
Updated: In the finite case (as in the reference), the only examples are $\mathrm{AGL}(1,q)$ for $q$ a prime power.


This follows, for example, by "Lucchini, Mainardis, Stellmacher, Transitive permutation groups with cyclic point stabilizers of maximum order. Geom. Dedicata 100 (2003), 117–121."


