Let \(H\) be a bialgebra over a field, and let \(B\) be both a left \(H\)-module algebra and a left \(H\)-comodule coalgebra. Then \(B\otimes H\) has the smash product structure of an algebra and the smash coproduct structure of a coalgebra (for a description of these structures, see R. Molnar [J. Algebra 47, 29-51 (1977; Zbl 0353.16004)]). The author investigates when this situation yields a bialgebra structure on \(B\otimes H\). In this case, he calls \((H,B)\) an admissible pair. He shows that this situation is characterized by the canonical algebra injections of \(B\) and \(H\) into \(B\otimes H\) and the canonical coalgebra projections of \(B\otimes H\) onto \(B\) and \(H\). For \((H,B)\) an admissible pair, he studies when \(B\otimes H\) is a Hopf algebra, the integrals of \(B\otimes H\), and when \(B\otimes H\) is semisimple or cosemisimple. He determines sufficient conditions for \((H,B)\) to be admissible when \(H\) is a Hopf algebra in terms of the canonical mappings involving \(H\) and \(B\otimes H\). Finally, he studies the special case where \(H\) is a group algebra and \(B\) is a group-like coalgebra on a group, and uses this to construct some new examples of semisimple cosemisimple involutory Hopf algebras.