Automorphism Groups of Countable Stable Structures

For every countable structure $M$ we construct an ${\mathrm{\aleph }}_0$-stable countable structure $N$ such that $Aut(M)$ and $Aut(N)$ are topologically isomorphic. This shows that it is impossible to detect any form of stability of a countable structure $M$ from the topological properties of the Polish group $Aut(M)$.