Let $X$ be a smooth closed subvariety of a complex abelian variety $A$. Assume $X$ is of general type and of codimension one with $\omega_X$ ample.

Often, people speak about the stabilizer $\mathrm{Stab}_A(X)$ of $X$ in $A$. This is the group of $a$ in $A$ such that $X+a = X$.

What is the relation of $\mathrm{Stab}_A(X)$ to $\mathrm{Aut}(X)$?

They are both finite. Are they equal? If the stabilizer is trivial, does that imply $\mathrm{Aut}(X)$ is trivial? What about vice versa? Does $\mathrm{Stab}_a(X)$ inject into $\mathrm{Aut}(X)$?

Crossposted from stackexchange, because I didn't get any replies there unfortunately: https://math.stackexchange.com/questions/4446811/difference-between-stabilizer-and-automorphism-group