  Let $\H_n$ be a (degenerate or non-degenerate) Hecke algebra of type
$G(\ell,1,n)$, defined over a commutative ring $R$ with one, and let $S(\bmu)$
be a Specht module for $\H_n$. This paper shows that the induced Specht module
$S(\bmu)\otimes_{\H_n}\H_{n+1}$ has an explicit Specht filtration.
