  We consider the following problem. Suppose $\alpha$ is an action of a locally
compact group $G$ on a $C^*$-algebra $A$, $H$ is a closed subgroup of $G$, and
$(\pi,U)$ is a covariant representation of $(A,H,\alpha)$. For which closed
subgroups $K$ containing $H$ is there a covariant representation $(\pi,V)$ of
$(A,K,\alpha)$ such that $V|_H=U$?
  We answer this problem by providing a criterion involving the induced
representation of $(A,G,\alpha)$. We then consider the dual problem for
coactions of locally compact groups.
