A permutation group $G \lt S_n$ is called generously transitive, if for each $i,j$ there exists a permutation that interchanges them. Is there a reasonable classification of such (finite) groups?
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I believe the answer is No, there is no good classification of these things. It might be helpful for you to know that a "generously transitive permutation group" is the same as a 2star transitive group. (See for instance the introduction to "The kstar Property for Permutation Groups" by Clough, Praeger, Schneider.) I can email you a copy of this paper if you need it. 


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. I've fixed the question. – Arturo Magidin Feb 28 '13 at 15:54