  The regular subgroup determining an induced Hopf Galois structure for a
Galois extension $L/K$ is obtained as the direct product of the corresponding
regular groups of the inducing subextensions. We describe here the associated
Hopf algebra and Hopf action of an induced structure and we prove that they are
obtained by tensoring the corresponding inducing objects. In order to deal with
their associated orders we develop a general method to compute bases and free
generators in terms of matrices coming from representation theory of Hopf
modules. In the case of an induced Hopf Galois structure it allows us to
decompose the associated order, assuming that inducing subextensions are
arithmetically disjoint.
