If an automorphism $\alpha$ of a C*-algebra $A$ is inner then whenever $A$ is a subalgebra of another C*-algebra $B$, $\alpha$ obviously extends to $B$.

Is the converse true: if an automorphism $\alpha$ of $A$ is such that whenever $A \subset B$ then $\alpha$ extends to an automorphism of $B$, is $\alpha$ necessarily inner?

The analogous question for groups has a positive solution, see: Are the inner automorphisms the only ones that extend to every overgroup?