There is a well known Morita equivalence between the group C*-algebra $C^*(H)$ and $C_0(G/H) \rtimes G$, where $H$ is a subgroup of $G$. The corresponding equivalence of representations is an incarnation of Mackey's imprimitivity theorem, see Rieffel's `Morita Equivalence for Operator Algebras', Example 1.

In topology, one can essentially treat a $H$-space $X$ as equivalent to the $G$ space $X \times_H G$. In particular, the $H$-equivariant K-theory of $X$ is isomorphic to the $G$-equivariant K-theory of $X \times_H G$. When $X$ is a point, this corresponds to the Morita equivalence of the previous paragraph.

My question is whether this Morita equivalence is a special case of an equivalence between $C_0(X) \rtimes H$ and $C_0(X \times_H G) \rtimes G$. When $X$ is a point, one uses $C^*(G)$ as the equivalence bimodule, but I don't know what to replace this with in the more general setting.

In case this is true: is there a general construction in $C^*$-algebras that is analogous to the construction of $X \times_H G$ from $X$. Explicitly, if $A$ is a $C^*$-algebra with $H$-action, is there a canonical $C^*$-algebra $\tilde A$ with $G$-action such that $A \rtimes H$ is Morita equivalent to $\tilde A \rtimes G$?