Let $A$ be a finite alphabet, $X$ = $(A^\mathbb{Z}, \sigma)$ the full shift, and $Y \subset X$ a subshift.

Question:Are there any general results characterizing whether automorphisms of $(Y, \sigma)$ are likely to extend to automorphisms of $X$ (that is, whether automorphisms of $(Y, \sigma)$ are likely to be restrictions of automorphisms of $X$).

I suspect that the answer is that automorphisms generally do not extend. For example, it may be the case that for most subshifts (for some appropriate meaning of "most"), the only automorphisms that extend are powers of $\sigma$.

Any thoughts or references would be greatly appreciated!