This paper studies 2- \((v,5,\lambda )\) designs admitting block-transitive automorphism groups \(S_c\wr S_d\) . We determine all such designs, and obtain that, if automorphism group of design is \(S_c\wr S_d\) , up to isomorphism, then there are 4 non-trivial 2- \((v,5,\lambda )\) designs with point-primitive automorphism group, and 4 non-trivial 2- \((v,5,\lambda )\) designs with point-imprimitive automorphism group.
