Doubly transitive automorphism groups of block designs

If G is a doubly transitive group of automorphisms of a block design with λ = 1, then for any block Δ of the design and any point α in Δ, the set Δ?{α} is a block of imprimitivity for G_α. What are sufficient conditions for a doubly transitive but not doubly primitive permutation group G to be a group of automorphisms of a non-trivial block design with λ = 1 ? Can the design or the group G be identified if there is a nonidentity automorphism in G fixing every point of some block of the design? Both of these questions are investigated and some answers are given.