  In a previous article, we defined a very flexible notion of suborbifold and
characterized those suborbifolds which can arise as the images of orbifold
embeddings. In particular, suborbifolds are images of orbifold embeddings
precisely when they are saturated and split. This article addresses the problem
of orbifold structure inheritance for three orbifolds
$\mathcal{Q}\subset\mathcal{P}\subset\mathcal{O}$. We identify an appealing but
ultimately inadequate notion of an inherited canonical orbifold substructure.
In particular, we give a concrete example where the orbifold structure of
$\mathcal{Q}$ is canonically inherited from $\mathcal{P}$, and the orbifold
structure of $\mathcal{P}$ is canonically inherited from $\mathcal{O}$, but the
orbifold structure of $\mathcal{Q}$ is not canonically inherited from
$\mathcal{O}$. On the other hand, it is easy to see that when $\mathcal{Q}$ is
embedded in $\mathcal{P}$, and $\mathcal{P}$ is embedded in $\mathcal{O}$, all
of the canonical inherited orbifold substructures will agree. We also
investigate the property of saturation in this context, and give an example of
a suborbifold with the canonical orbifold substructure that is not saturated.
