We consider a new construction of automorphic representations of GL2(A), using the general idea of automorphic descent methods. In particular, four families of examples are considered. The representations τ of GL2(A) are obtained from automorphic representations π on reductive groups H, which contain a reductive subgroup G of GSp2n. We give criteria for nonvanishing and for cuspidality of the constructed representations τ, in terms of periods or co-periods and by the holomorphy at s=1 of certain l-functions of π. We also calculate the relation of the unramified parameters in certain cases, which indicates that τ and π fit partially to the Langlands functorial principle. We prove some low-rank cases to support our conjectures.