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This is a follow-up to a paper with the same title and by the same authors.In that paper, all groups were assumed to be abelian, and we are now aiming togeneralize the results to nonabelian groups.  The motivating point is Pedersen's theorem, which does hold for an arbitrarylocally compact group $G$, saying that two actions $(A,\alpha)$ and $(B,\beta)$of $G$ are outer conjugate if and only if the dual coactions$(A\rtimes_{\alpha}G,\widehat\alpha)$ and $(B\rtimes_{\beta}G,\widehat\beta)$of $G$ are conjugate via an isomorphism that maps the image of $A$ onto theimage of $B$ (inside the multiplier algebras of the respective crossedproducts).  We do not know of any examples of a pair of non-outer-conjugate actions suchthat their dual coactions are conjugate, and our interest is thereforeexploring the necessity of latter condition involving the images, and we havedecided to use the term "Pedersen rigid" for cases where this condition isindeed redundant.  There is also a related problem, concerning the possibility of a so-calledequivariant coaction having a unique generalized fixed-point algebra, that wecall "fixed-point rigidity". In particular, if the dual coaction of an actionis fixed-point rigid, then the action itself is Pedersen rigid, and no exampleof non-fixed-point-rigid coaction is known.

Rigidity theory for $C^*$-dynamical systems and the "Pedersen Rigidity Problem", II