Let $\mathfrak{M}$ be a countable saturated structure, and assume that $D(\nu)$ is a strongly minimal formula (without parameter) such that $\mathfrak{M}$ is the algebraic closure of $D(\mathfrak{M})$. We will prove the two following theorems: Theorem 1. If $G$ is a subgroup of $\operatorname{Aut}(\mathfrak{M})$ of countable index, there exists a finite set $A$ in $\mathfrak{M}$ such that every $A$-strong automorphism is in $G$. Theorem 2. Assume that $G$ is a normal subgroup of $\operatorname{Aut}(\mathfrak{M})$ containing an element $g$ such that for all $n$ there exists $X \subseteq D(\mathfrak{M})$ such that $\operatorname{Dim}(g(X)/X) > n$. Then every strong automorphism is in $G$.