Transitive groups of degree \(2p+k\) (Q1127979)

Let \(p\) be an odd prime and let \(G\) be a transitive permutation group of degree \(2p+k\) where \(k\in\{1,2,p\}\). Conditions are investigated which allow the conclusion that \(G\) is doubly transitive. E.g. a group \(G\) of degree \(2p+1\) and even order is doubly transitive iff \(G\) has an element of order \(p\) and degree \(2p\), or a primitive group \(G\) of even order and degree \(3p\geq 15\) is doubly transitive if \(p\) is not of the form \(3a^2+3a+1\) or \((3b^2+3b+2)/4\) and the rank of \(G\) less than \(4\).