  Let $G$ be a locally compact abelian group. By modifying a theorem of
Pedersen, it follows that actions of $G$ on $C^*$-algebras $A$ and $B$ are
outer conjugate if and only if there is an isomorphism of the crossed products
that is equivariant for the dual actions and preserves the images of $A$ and
$B$ in the multiplier algebras of the crossed products. The rigidity problem
discussed in this paper deals with the necessity of the last condition
concerning the images of $A$ and $B$.
  There is an alternative formulation of the problem: an action of the dual
group $\hat G$ together with a suitably equivariant unitary homomorphism of $G$
give rise to a generalized fixed-point algebra via Landstad's theorem, and a
problem related to the above is to produce an action of $\hat G$ and two such
equivariant unitary homomorphisms of $G$ that give distinct generalized
fixed-point algebras.
  We present several situations where the condition on the images of $A$ and
$B$ is redundant, and where having distinct generalized fixed-point algebras is
impossible. For example, if $G$ is discrete, this will be the case for all
actions of $G$.
