This is a survey article on barely transitive groups. It also involves some recent results in the case of a non-locally finite barely transitive group.
Locally finite group, barely transitive group, infinite permutation group
KUZUCUOĞLU, MAHMUT (2007) "Barely Transitive Groups," Turkish Journal of Mathematics: Vol. 31: No. 5, Article 6. Available at: https://journals.tubitak.gov.tr/math/vol31/iss5/6