  Let $G$ be a countable discrete amenable group, ${\cal M}$ a McDuff factor
von Neumann algebra, and $A$ a separable nuclear weakly dense C$^*$-subalgebra
of ${\cal M}$. We show that if two centrally free actions of $G$ on ${\cal M}$
differ up to approximately inner automorphisms then they are outer conjugate by
an approximately inner automorphism, in the operator norm topology, which makes
$A$ invariant. In addition, when $A$ is unital, simple, and with a unique
tracial state and $\alpha$ is an automorphism of $A$ we also show that the
aperiodicity of $\alpha$ on the von Neumann algebra is equivalent to the weak
Rohlin property.
