Homomorphic extensions of groups of diffeomorphisms (Q675329)

For submanifolds \(N\) of smooth manifolds \(M\) the question whether a (continuous) group homomorphism \(\sigma:\text{Diff}(N)\to\text{Diff}(M)\) can be found such that \(\sigma(f)|_N=f\) is treated in some examples. In particular for \(S^1\subset D^2\) and \(\partial M\subseteq M:=\mathbb{R}\times[0,1]\) the answer is shown to be negative, and is positive for some subgroups of \(\text{Diff}(\partial M)\) in the second example.