I think this is probably a very simple question, but I've been puzzling over it for a while and can't seem to get anywhere.
Suppose $M$ is a structure, $\alpha$ is an automorphism of $M$, and $N$ is an elementary extension of $M$. Does $\alpha$ necessarily extend to an automorphism of $N$?
It seems to me that the answer should be no, but I can't construct a counterexample. In case the answer is no, is there some natural condition on $N$ (or on how $M$ sits inside $N$) that guarantees that each automorphism of $M$ extends to an automorphism of $N$?