Subsets of $\mathbb{S}^n$ fixed by an orientation-reversing self-homeomorphism — Part 2 This is another special case of this question.
Recall that we call a subset  $Z$ of $\mathbb{S}^n$ ambiently-reversible, if there is an orientation-reversing homeomorphism $h: \mathbb{S}^n \to \mathbb{S}^n$ fixing $Z$ pointwise.
Question 1: Is there an ambiently-reversible wild Cantor set in $\mathbb{S}^n, n\geq 3$? Is there a non-ambiently-reversible one?
 A: Similar to the proof of the exercise problem in Part 1 of the OP Question, one may generalize it to higher dimensions. However, since the direct extension of Jordan-Schoenflies theorem is impossible, the generalization is not obvious. I think the OP is related to the problem of exotic involution of $S^n$. When $Z$ is a Cantor set, $Z$ can be considered as a closed subset of a wildly embedded sphere in $S^3$. the Alexander horned sphere $\Sigma$. $\Sigma$ separates $S^3$ and the closures of the corresponding components in the complement are crumpled 3-cubes. Denote the crumpled 3-cube by $AH$. As I made in the comment, R. H. Bing stunned the community by showing that $AH \cup_{id} AH$ $\approx S^3$, the argument relies on his shrinking criterion. The point is that his result shows that $S^3$ admits an involution fixing a wild 2-sphere, the Alexander horned sphere $\Sigma$. Note that $Z \subset \Sigma$. Then $Z$ is ambiently-reversible.
This might be a rudimentary example to the OP. Because it seems that the OP requires that the Cantor set $Z$ is wild in $S^3$. The one in the Alexander horned sphere is not wild in $S^3$. Indeed, $Z$ is twice flat in the sense that it is flat both as a subset of the Alexander horned sphere and as a subset of $S^3$. However, there are results may handle Cantor sets which are wildly embedded in the ambient space. For instance, for any crumpled $n$-cube $C$, $n \geq 5$, $C ∪_{Id} C \approx S^n$ iff $C$ satisfies the Disjoint Disks Property. See Thm 7.10.6, P. 413, Daverman-Venema.
Using the involution of spheres alone might not answer whether there exists a non-ambiently-reversible Cantor set, a potential example is produced by Daverman. Utilizing a sticky Cantor set in $S^n$ ($n\geq 4$) constructed by Krushkal, he showed that the sewings of some crumpled cubes $C \cup_{Id} C$ is not homeomorphic to sphere. Furthermore, there is no homeo $H: Bd C \to Bd C$ close to $Id$ such that $C \cup_{H} C \approx S^n$.
