Let G be a finite group and let K be a hilbertian field. Many finite groups have been shown to satisfy the
arithmetic lifting property over K, that is, every G-Galois extension of K arises as a specialization of a
geometric branched covering of the projective line defined over K. The paper explores the situation when
a semidirect product of two groups has this property. In particular, it shows that if H is a group that
satisfies the arithmetic lifting property over K and A is a finite cyclic group
then G = A [rtimes ] H also satisfies
the arithmetic lifting property assuming the orders of H and A are relatively prime and the characteristic
of K does not divide the order of A. In this case, an arithmetic lifting for any A[wreath ]H-Galois extension
of K is explicitly constructed and the existence of the arithmetic lifting for any G-Galois extension is
deduced. It is also shown that if A is any abelian group, and H is the group with the arithmetic lifting
property then A[wreath ]H satisfies the property as well, with some assumptions on the ground field K. In the
construction properties of Hilbert sets in hilbertian fields and spectral sequences in étale cohomology
are used.