Say, we have a countable ICC group G, a Hilbert space with a basis indexed by the group elements, the group algebra generated by the left regular representation of G on this Hilbert space, and its norm and weak closures, the reduced C-* and the von Neumann algebras. On the other hand, an automorphism of G also permutes the basis of H (when applied to the indices) and thus defines a unitary on H. What's the relationship of this unitary to the group algebras? Is it ever in, say, the von Neumann group factor? How do the properties of this unitary reflect whether it comes from an inner group automorphism or not?
These unitaries, representing automorphisms of G, also give rise to an operator algebra. How is this algebra related to the group's, say, von Neumann algebra?

Say, we have a countable ICC group $G$, a Hilbert space $H$ with a basis indexed by the group elements, the group algebra generated by the left regular representation of $G$ on this Hilbert space, and its norm and weak closures, the reduced C-* and the von Neumann algebras. On the other hand, an automorphism of $G$ also permutes the basis of $H$ (when applied to the indices) and thus defines a unitary on $H$. What's the relationship of this unitary to the group algebras? Is it ever in, say, the von Neumann group factor? How do the properties of this unitary reflect whether it comes from an inner group automorphism or not?
These unitaries, representing automorphisms of $G$, also give rise to an operator algebra. How is this algebra related to the group's, say, von Neumann algebra?