Transitive permutation groups in which all derangements are involutions

Let GG be a transitive permutation group in which all derangements are involutions. We prove that GG is either an elementary abelian 2-group or is a Frobenius group having an elementary abelian 2-group as kernel. We also consider the analogous problem for abstract groups, and we classify groups GG with a proper subgroup HH such that every element of GG not conjugate to an element of HH is an involution.