Analyzing the answer of @James Hanson to my preceding question, I realized that this question also has a simple negative answer: the quotient space $c\mathbb N:=\beta\mathbb N/\{p,q\}$ of $\beta\mathbb N$ by any doubleton $\{p,q\}\subset\beta\mathbb N\setminus\mathbb N$ is not homeomorphic to $\beta\mathbb N$ but has the smallest possible permutation group $S_{\mathbb N,c\mathbb N}=S_{<\mathbb N}$.

Analyzing the answer of @James Hanson to my preceding question, I realized that this question also has a simple negative answer: the quotient space $c\mathbb N:=\beta\mathbb N/\{p,q\}$ of $\beta\mathbb N$ by any doubleton $\{p,q\}\subset\beta\mathbb N\setminus\mathbb N$ is not homeomorphic to $\beta\mathbb N$ but has the smallest possible permutation group $S_{\mathbb N,c\mathbb N}=S_{<\mathbb N}$. This compactification $c\mathbb N$ also is not soft (according to this definition).