The following is supposed to be "clear" according to Kueker, but I could not see why. Can anyone help?

Let $A$ be a countable structure with uncountable many automorphisms. Then for every $\vec{a}\in A^{<\omega}$, $(A,\vec{a})$ has a non-trivial automorphism, i.e. there exists some $f:A\rightarrow A$ such that $f\neq id$ and $f(a)=a$, for all $a\in \vec{a}$.

Note: The argument is part of a larger proof from Infinitary Logic: In memoriam Carol Karp