For a compactification $c\mathbb N$ of $\mathbb N$ let $\mathcal H(c\mathbb N,\mathbb N)$ be the group of homeomorphisms $h:c\mathbb N\to c\mathbb N$ such that $h(x)=x$ for all $x\in c\mathbb N\setminus\mathbb N$. The group $\mathcal H(c\mathbb N,\mathbb N)$ determines the subgroup $$S_{\mathbb N,c\mathbb N}:=\{h{\restriction}\mathbb N:h\in\mathcal H(c\mathbb N,\mathbb N)\}$$ in the permutation group $S_{\mathbb N}$ of $\mathbb N$.
Observe that for the one-point compactification $\alpha\mathbb N$ of $\mathbb N$ the group $S_{\mathbb N,\alpha\mathbb N}$ coincides with the whole group $S_{\mathbb N}$ whereas for the Stone-Cech compactification $\beta\mathbb N$ of $\mathbb N$ the group $S_{\mathbb N,\beta\mathbb N}$ coincides with the group $S_{<\mathbb N}$ of finitely supported permutations of $\mathbb N$ (i.e., permutations that move only finitely many points of $\mathbb N$).
For any compactification $c\mathbb N$ of $\mathbb N$ we have $$S_{<\mathbb N}=S_{\mathbb N,\beta\mathbb N}\subset S_{\mathbb N,c\mathbb N}\subset S_{\mathbb N,\alpha\mathbb N}=S_{\mathbb N}$$so $S_{\mathbb N,c\mathbb N}$ is intermediate between $S_{\mathbb N,\beta\mathbb N}$ and $S_{\mathbb N,\alpha\mathbb N}$.
It is easy to prove that the one-point compactification of $\mathbb N$ is the unique compactification $c\mathbb N$ of $\mathbb N$ with $S_{\mathbb N,c\mathbb N}=S_{\mathbb N}$.
Question. Is the Stone-Cech compactification a unique compactification of $\mathbb N$ with $S_{\mathbb N,c\mathbb N}=S_{<\mathbb N}$?
Remark. Negative answer to this question will imply negative answer to this MO-problem.
