Let (D,B) be an admissible pair. Then recall that $B;{ imes}^L_HD^{{ ightarrow}{pi}_D}_{{leftarrow}i_D};D$ are bialgebra maps satisfying ${pi}_D{circ}i_D=I$. We have solved a converse in case D is a Hopf algebra. Let D be a Hopf algebra with antipode $S_D$ and be a left H-comodule algebra and a left H-module coalgebra over a field $k$. Let A be a bialgebra over $k$. Suppose $A^{{ ightarrow}{pi}}_{{leftarrow}i}D$ are bialgebra maps satisfying ${pi}{circ}i=I_D$. Set ${Pi}=I_D*(i{circ}s_D{circ}{pi}),B=Pi(A)$ and $j:B{ ightarrow}A$ be the inclusion. Suppose that ${Pi}$ is an algebra map. We show that (D,B) is an admissible pair and $B^{leftarrow{Pi}}_{ ightarrow{j}}A^{ ightarrow{pi}}_{leftarrow{i}}D$ is an admissible mapping system and that the generalized biproduct bialgebra $B{ imes}^L_HD$ is isomorphic to A as bialgebras.