Let $A$ be a $C^*$-algebra. The group $Aut(A)$ of $\ast$-automorphisms of $A$ is usually equipped either with the pointwise norm topology, i.e. the topology generated by the semi-norms $\lVert \varphi \rVert_a = \lVert \varphi(a) \rVert$ or with the uniform topology, i.e. the one which it earns by inclusion into the Banach space of bounded linear operators from $A$ to $A$. Moreover, let $U(A) = U(M(A))$ be the unitary group of the multiplier algebra equipped with the strict topology. Now, the map $Ad : U(A) \to Aut(A)$ is continuous if $Aut(A)$ carries the pointwise norm topology. It induces a continuous bijection $U(A) / Z(U(A)) \to Inn(A)$ onto the inner automorphisms, which is not a homeomorphism unless the $C^\ast$-algebra is a continuous trace $C^{\ast}$-algebra. So, my questions are

Is the induced bijection $U(A)/Z(U(A)) \to Inn(A)$ a homeomorphism, if $Aut(A)$ carries the uniform topology?

Is there a natural topology on $Aut(A)$, which induces a homeomorphism $U(A)/Z(U(A)) \to Inn(A)$?