Regular semidirect products of semigroups have been studied by G. B. Preston [5], orthodox by T. Saito [6] and $E$-inversive by F. Cabro and M. Micoli [3] e.t.c. In the present paper we study semidirect products belonging to one of the following classes of semigroups: $\pi$-regular, semilattices of Archimedean semigroups, Archimedean, left Archimedean, right Archimedean and other. At the end of the paper we give a new proof of the Wilkinsons theorem.