Here is another problem in "Model Theory" (Chang-Keisler 2012) that I need help.
Exercise $1.3.22$. Let $\mathfrak{A}$ be a countable model for a countable language. Prove that if the simple expansion $(\mathfrak{A},$ $b)$ has more than one automorphism for each finite sequence $b$ of elements of $A$, then $\mathfrak{A}$ has $2^{\omega}$ automorphisms.
I know how to prove automorphism by back and forth method. But I am not sure how to prove the number of automorphisms for a model.
The idea is the following (I come short of proving that the construction works as intended).
Construct a function $b:2^{<\omega}\to A$ such that
$b(\alpha^\frown0)\neq b(\alpha^\frown1)$ for all $\alpha\in2^{<\omega}$;
every infinite branch enumerates $A$, i.e. $\{b(\alpha{\restriction} n) : n\in\omega\}$ is an enumeration of $A$ for all $\alpha\in2^{\omega}$;
there is an automorphism $f_{\alpha,\beta}: b(\alpha{\restriction} i)\mapsto b(\beta{\restriction} i)$ for every $n$, every $\alpha,\beta\in2^{n}$ and $i<n$.
Assume inductively that $b(\alpha)$ is defined for all $\alpha\in2^{<m}$ and that (2) holds for $n<m$.
Pick any $\gamma\in2^{m-1}$. By assumption, there are two distinct $b(\gamma^\frown0), b(\gamma^\frown 1)$ that are conjugate over $\{b(\gamma{\restriction}i)\, :\, i<n\}$.
Now for every $\alpha\in2^{m-1}$ let $f_{\gamma,\alpha}$ be the automorphism obtained from (2). Define
$\quad b(\alpha^\frown0)=f_{\gamma,\alpha}\big(b(\gamma^\frown0)\big)$ and
$\quad b(\alpha^\frown1)=f_{\gamma,\alpha}\big(b((\gamma^\frown1)\big)$.