11.69 (1990)

A group $G$ acting on a set $\Omega$ will be said to be $1$-$\{2\}$-transitive if it acts transitively on the set $\Omega^{1,\{2\}} = \{(\alpha, \{\beta, \gamma\}) \mid \alpha, \beta, \gamma$ distinct$\}$. Thus $G$ is $1$-$\{2\}$-transitive if and only if it is transitive and a stabilizer $G_\alpha$ is 2-homogeneous on $\Omega \setminus \{\alpha\}$. The problem is to classify all (infinite) permutation groups that are $1$-$\{2\}$-transitive but not 3-transitive.