An action $\alpha$ of a locally compact topological group G on a unital $C^*$-algebra $A$ is called inner if there exists a continuous group homomorphism $u\colon G\to U(A)$ such that $\alpha_g(a)=u_gau_g^*$ for all $g\in G$ and all $a\in A$.
Question: let $\alpha\colon S^1\to Aut(M_n)$ be a continuous action. Does it follow that $\alpha$ is inner?
For every $\zeta\in S^1$, the automorphism $\alpha_\zeta$ is inner, say $\alpha_\zeta=Ad(u_\zeta)$ for some unitary $u_\zeta$ in $M_n$. This unitary is unique up to multiplication by elements of $S^1\cdot 1_n\subseteq M_n$ (the unitaries in the center of $M_n$). Is it known whether one can choose the map $\zeta\mapsto u_\zeta$ to be a continuous group homomorphism?
I suppose this may be known to some people but I could not find any references.