Let $A$ be a (semi-)abelian variety over an algebraically closed field $K$, and $X$ be a closed irreducible subvariety. Can $X$ have a non-trivial finite stabilizer ? By stabilizer, I mean the closed subgroup $S\subset A$ such that $X$ is stable by translation by any point in $S$. This question is equivalent to the one in the title since you can always take the quotient by the neutral component of $S$ to reduce to a finite group.
No. Take a curve in its Jacobian and pull it back by multiplication by some n. The resulting pullback is a curve invariant by the n torsion.