Suppose that $A$ is a formal power series ring over $k$, where $k$ is an algebriac closed field. $I,J$ are ideals of $A$.
If $\phi:A/I\rightarrow A/J$ is a surjection. Can we induce a $k-$algebra automorphism of $A$ by $\phi$ to make the corresponding diagram commutative?
So it’s equivalent to following questiion:
If $\psi:A\rightarrow A/I$ is a surjection(may not be canonical).
Can we induce a $k-$algebra automorphism of $A$?
If $I$ is contained in the square of the maximal ideal, any lift will give you an automorphism of $A$. This is also true even if $I$ contains a variable (but no longer any lift) and I will let you think about it.