Given finite subset of $\mathbb C[[X]]$, does there exist a ring automorphism of $\mathbb C[[X]]$, fixing $A$, but not fixing $\mathbb C[X]$ set wise? For every  finite subset $A $ of $\mathbb C[[X]]$, does there exist a ring automorphism (bijective ring endomorphism ) $f$ of $\mathbb C[[X]]$ such that $f(a)=a, \forall a \in A$ but $f(\mathbb C[X]) \ne \mathbb C[X]$ ? 
Related Are $\mathbb C$ , $\mathbb C[X]$ definable in $\mathbb C[[X]]$?  because if $D \subseteq M$ is definable by a set $A \subseteq M$  then $f(D)=D, \forall f \in Aut_A M$ , and in the related case we mean definable by some finite subset of $M$. 
 A: I can't answer this question, but I can reduce it to your other question. 
Claim: The following are equivalent. 


*

*There exists a finite set $A\subseteq \mathbb{C}[[X]]$ such that every automorphism of $\mathbb{C}[[X]]$ which fixes $A$ pointwise fixes $\mathbb{C}$ setwise.

*There exists a finite set $B\subseteq \mathbb{C}[[X]]$ such that every automorphism of $\mathbb{C}[[X]]$ which fixes $B$ pointwise fixes $\mathbb{C}[X]$ setwise.
Proof: (1)$\implies$(2). Let $B = A\cup \{X\}$. Then for any automorphism $f$ which fixes $B$ pointwise, $f$ fixes $\mathbb{C}$ setwise and $f(X) = X$, so for any $a_nX^n+\dots+a_1X + a_0\in \mathbb{C}[X]$, $$f(a_nX^n + \dots + a_1X + a_0) = f(a_n)X^n + \dots + f(a_1)X + f(a_0)\in \mathbb{C}[X].$$
(2)$\implies$(1). Let $A = B$. Then any automorphism $f$ which fixes $A$ pointwise fixes the subring $\mathbb{C}[X]$ setwise, so it restricts to an automorphism of $\mathbb{C}[X]$. In particular, it fixes $\mathbb{C} = 0\cup \mathbb{C}^\times$ setwise, since $\mathbb{C}^\times$ is the group of units of $\mathbb{C}[X]$.
