If $A$ is a local artinian $k$-algebra, $X_1,X_2$ are finite type schemes flat over $A$, and $f:X_1\rightarrow X_2$ is a morphism over $A$ that is an isomorphism after restricting to closed fibers, then $f$ is an isomorphism.
This was proved here in the case that $X_1$ and $X_2$ are affine.
I am interested in automorphisms of $X$ over $A$ which we know to be a subset of the group of endomorphism of $X$ over $A$ which restrict to automorphisms of the fiber $X_0$.
Let $Aut(X/X_0)$ represent the group of automorphism of $X$ which restrict to the identity on $X_0$.
It seems like I should be able to act on $Aut(X/X_0)$ with $Aut(X_0)$ to recover $Aut(X)$.
Is this ever the case?
Here I am assuming that I have a local description of elements in $Aut(X/X_0)$ in terms of local coordinates on $X_0$. It seems like I could just pullback the automorphism in $Aut(X/X_0)$ via automorphism in $Aut(X_0)$.