Assume we are in the complex setting. Let $X$ be a surface, $C$ a curve on $X$. Say $X-C$ is isomorphic to some $X'-C'$ whith $X'$ a surface and $C'$ a curve on $X'$. If it helps we may assume that $X$ and $X'$ is fibered over $\mathbb P^1$ and $C$, $C'$ are fibers in the respective fibrations.
The question is: If I have an automorphism on $X$ that leaves $C$ invariant, does there exist an induced automorphism on $X'$ (that would automatically leave $C'$ invariant)? If not in general, are there conditions for this to hold?