While the answers by Eric and ARupinski give negative examples for your question, here is the precise characterization for when the answer is yes: Let $\alpha$ be an automorphism of the transitive subgroup $G\le S_n$, and $G_1$ be the stabilizer of $1$ in $G$. Then $\alpha$ extends to an automorphism of $S_n$ if and only if $G_1$ and $\alpha(G_1)$ are conjugate in $S_n$.

The necessity of the condition is clear, and the sufficiency is a nice exercise. I believe the result is also in the permutation groups book by Dixon-Mortimer.

While the answers by Eric and ARupinski give negative examples for your question, here is the precise characterization for when the answer is yes: Let $\alpha$ be an automorphism of the transitive subgroup $G\le S_n$, and $G_1$ be the stabilizer of $1$ in $G$. Then $\alpha$ extends to an inner automorphism of $S_n$ if and only if $G_1$ and $\alpha(G_1)$ are conjugate in $S_n$.

The necessity of the condition is clear, and the sufficiency is a nice exercise. I believe the result is also in the permutation groups book by Dixon-Mortimer.