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?
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. 


Clearly every generously transitive permutation group $G < {\rm S}_n$ acts primitively on $\{1,\dots,n\}$. Therefore we can find all generously transitive permutation groups of degree $n$ by searching through the primitive groups of degree $n$. The latter are available in the Primitive Permutation Groups Library of GAP, for all $n \leq 2500$. The following GAP function tests whether a given group is generously transitive:
Now let's check how many of the primitive groups of degree $\leq 100$ are even generously transitive:
Hence over $90$ percent of the primitive groups of degree $\leq 100$ are even generously transitive.  Therefore, perhaps rather than asking for a classification of generously transitive groups, one might ask for a classification of those primitive permutation groups which are not generously transitive. 


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