I believe that the following statement is true:
Let $U,V\subset \mathbb{R}^n$ be open sets, $K\subset U$ compact, and $\gamma:U\to V$ a diffeomorphism. Then there is a diffeomorphism $\Gamma:\mathbb{R}^n\to \mathbb{R}^n$ such that $\Gamma|_K=\gamma|_K$.
However, I do not know how to prove it in a few lines, and I do not know a reference for it. If someone told me a resource where the statement is proved, it would be really helpful.
Of course, if the statement is actually wrong, then it would be even more helpful if someone told me.
Thanks in advance!
Edit: As shown below, the statement is not true as stated above. However, the special case of contractible (or say, just simply connected) $K$ ist also of interest to me. What can we say then?
As stated it is false.
Let $$ U = (-1,0) \cup (10,11) \subset \mathbb{R}$$ and $$ V = (1,2) \cup (15,16) \subset \mathbb{R} $$ Take $$ K = \{ -2/3, -1/3 , 10.5\} $$
Let $\gamma$ be the map $$ \gamma(x) = \begin{cases} x + 16 & x \in (-1,0) \\ x - 9 & x \in (10,11)\end{cases} $$ which is clearly a diffeomorphism of $U$ and $V$.
But $\gamma|_{K}$ is not order preserving, so cannot equal the restriction of a diffeomorphism of $\mathbb{R}$ to itself.
For a connected example: let $r = \sqrt{x^2 + y^2}$ on $\mathbb{R}^2$. Let $U = V = \{r \in (1/2,2) \}$. Let $K = \{r \in [3/4,4/3]\}$. Let $\gamma$ be the inversion around the unit sphere: $(r,\theta) \mapsto (1/r,\theta)$. This is a diffeomorphism of $U$ to $V$. And also an automorphism of $K$. But any diffeomorphism extension of $\gamma|_K$ need to give a diffeomorphism betweeh the $B(3/4)$ and $\overline{B(4/3)}^C$ which is not possible since the two are not even homeomorphic.