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?
This is by no means a full answer, but Kishimoto has shown in Theorem 4.1 of his paper "Universally weakly inner one-parameter automorphism groups" that for an automorphism $\alpha$ of a separable $C^*$-algebra the following statements are equivalent:
(1) $\alpha$ is extendible in every irreducible representation
(2) $\alpha$ is universally weakly inner
If I read the last property correctly, then this means that there is a $u \in A^{**}$ such that $\alpha(a) = uau^*$.
