Suppose we have a diffeomorphism $f:{\mathbb{S}}^n_{+}\to\mathbb{S}^n$ of class $C^1$ of the closed upper hemisphere onto a submanifold of $\mathbb{S}^n$ with boundary.

Question.Is it possible to extend it to a diffeomorphism $F:\mathbb{S}^n\to\mathbb{S}^n$?

I believe that when $n=2$ this should follow from the smooth Schoenflies theorem, but still some work is necessary. I think I know how to do, but I did not write a rigorous proof.

From what I understand when $n\geq 3$, the generalized Schoenflies theorem allows us to extend $f$ to a homeomorphism, but that is much less than extending to a diffeomorphism.

I expect that the answer might depend on $n$ ($n\geq 3$, $n=4$, $n\geq 5$).

I would greatly appreciate it if you could provide references where I cold find relevant theorems (if there are any), including the case $n=2$. I need references for a proper citation in my paper.

**Edit:** This question is strictly related to another post: Gluing two diffeomorphisms together

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