  We give a modular branching rule for certain wreath products as a
generalization of Kleshchev's modular branching rule for the symmetric groups.
Our result contains a modular branching rule for the complex reflection groups
$G(m,1,n)$ (which are often called the generalized symmetric groups) in
splitting fields for $\mathbb{Z}/m\mathbb{Z}$. Especially for $m=2$ (which is
the case of the Weyl groups of type $B$), we can give a modular branching rule
in any field. Our proof is elementary in that it is essentially a combination
of Frobenius reciprocity, Mackey theorem, Clifford's theory and Kleshchev's
modular branching rule.
