In the paper "Orbit Equivalence and Topological Conjugacy of Affine Actions on Compact Abelian Groups", S. Bhattacharya shows (Theorem 3) the folowwing:

Theorem. Given two actions $\alpha$ and $\beta$ of a discrete group $\Gamma$ on a compact connected metrizable abelian group $K$ by continuous group automorphisms the following are equivalent:

1. there exists a homeomorphism $F\colon K \to K$ such that $\alpha_\gamma= F\beta_\gamma F^{-1}$ for every $\gamma\in \Gamma$.

2. there exists a continuous group automorphism $F\colon K \to K$ such that $\alpha_\gamma = F\beta_\gamma F^{-1}$ for every $\gamma \in \Gamma$.

By passing to the Pontryagin dual this becomes a statement about discrete countable torsion-free abelian groups. My question concerns the generalization to non-abelian groups.

Question. Given two actions $\alpha$ and $\beta$ of a discrete group $\Gamma$ on a discrete countable torsion-free group $G$ by group automorphisms, are the following equivalent?

1. there exists an automorphism of the reduced $C^*$-algebra of $G$ which conjugates the actions induced by $\alpha$ and $\beta$

2. there exists an automorphism of $G$ which conjugates $\alpha$ and $\beta$.

This question is also interesting when we restrict the attention fo $\Gamma = \mathbb Z$ (i.e. to pairs of automorphisms.)

In the paper "Orbit Equivalence and Topological Conjugacy of Affine Actions on Compact Abelian Groups", S. Bhattacharya shows (Theorem 3) the following:

Theorem. Given two actions $\alpha$ and $\beta$ of a discrete group $\Gamma$ on a compact connected metrizable abelian group $K$ by continuous group automorphisms the following are equivalent:

1. there exists a homeomorphism $F\colon K \to K$ such that $\alpha_\gamma= F\beta_\gamma F^{-1}$ for every $\gamma\in \Gamma$.

2. there exists a continuous group automorphism $F\colon K \to K$ such that $\alpha_\gamma = F\beta_\gamma F^{-1}$ for every $\gamma \in \Gamma$.

By passing to the Pontryagin dual this becomes a statement about discrete countable torsion-free abelian groups. My question concerns the generalization to non-abelian groups.

Question. Given two actions $\alpha$ and $\beta$ of a discrete group $\Gamma$ on a discrete countable torsion-free group $G$ by group automorphisms, are the following equivalent?

1. there exists an automorphism of the reduced $C^*$-algebra of $G$ which conjugates the actions induced by $\alpha$ and $\beta$

2. there exists an automorphism of $G$ which conjugates $\alpha$ and $\beta$.

This question is also interesting when we restrict the attention fo $\Gamma = \mathbb Z$ (i.e. to pairs of automorphisms.)