Here is a proof that $C(X)\rtimes H$ is Morita equivalent to $C(X\times_H G)\rtimes G$. I'm assuming here that $G,H$ are countable discrete groups and $X$ is second countable. First of all, since I prefer to work with left actions, I will assume that $H$ acts on the left of $X$ and I will write instead $G\times_H X$ for $(G\times X)/H$ where the action of $H$ is on the right via $(g,x)h = (gh,h^{-1}x)$. I hope that is fine.

I will show that the corresponding transformation etale groupoids are Morita equivalent groupoids. See https://www.aidansims.com/files/GroupoidsNotes-2017.pdf section 3.4. It is a well-known result of Renault that Morita equivalent Hausdorff groupoids have strongly Morita equivalent $C^*$-algebras and the proof of Theorem 3.4.4 in the above reference tells you how to build the imprimitivity bimodule from the groupoid equivalence.

I will write $g\otimes x$ to denote the class of $(g,x)$ in $G\times_H X$ since this is like a tensor product in that $gh\otimes x=g\otimes hx$ for $h\in H$. Then $H\ltimes X$ is the groupoid with unit space $X$ and arrow space $H\times X$ where the arrow $(h,x)\colon x\to hx$. The product is $(h,h'x)(h',x)= (hh',x)$ and the topology is the product topology on $H\times X$ and the usual topology on $X$. The inverse is $(h,x)^{-1}=(h^{-1},hx)$. The groupoid $G\ltimes (G\times_H X)$ is defined similarly but has object space $G\times_H X$ and arrow space $G\times (G\times_H X)$.

To find a Morita equivalence I need a principal bibundle for these groupoids with the appropriate properties. The precise definition is on page 23 of the linked file.

Let $Z=G\times X$ with the product topology. We have open maps $p\colon Z\to G\times_H X$ and $q\colon Z\to X$ given by the quotient map in the first case and the projection in the second case. These can be use as anchors (or moment maps) for a left action of $G\ltimes (G\times_H X)$ and a right action of $H\ltimes X$ which commute. The left action of $G\ltimes (G\times H_X)$ is given by $(g_1,g_0\otimes x)(g_0,x) = (g_1g_0, x)$ and the right action of $H\ltimes X$ is given by $(g_0,x)(h,h^{-1}x) = (g_0h,h^{-1}x)$. It is easy to check that these are free and proper commuting actions. Also the quotient of $Z$ by the action of $G\ltimes (G\times_H X)$ is homeomorphic to $X$ via $q$ and the quotient of $Z$ by the action of $H\ltimes X$ is $G\times_H X$ by construction. Thus this bibundle gives a Morita equivalence of groupoids and hence a Morita equivalence of $C^*$-algebras.

Here is a proof that $C(X)\rtimes H$ is Morita equivalent to $C(X\times_H G)\rtimes G$. I'm assuming here that $G,H$ are countable discrete groups and $X$ is second countable. First of all, since I prefer to work with left actions, I will assume that $H$ acts on the left of $X$ and I will write instead $G\times_H X$ for $(G\times X)/H$ where the action of $H$ is on the right via $(g,x)h = (gh,h^{-1}x)$. I hope that is fine.

I will show that the corresponding transformation etale groupoids are Morita equivalent groupoids. See Sims - Hausdorff étale groupoids and their $C^*$ algebras section 3.4. It is a well-known result of Renault that Morita equivalent Hausdorff groupoids have strongly Morita equivalent $C^*$-algebras and the proof of Theorem 3.4.4 in the above reference tells you how to build the imprimitivity bimodule from the groupoid equivalence.

I will write $g\otimes x$ to denote the class of $(g,x)$ in $G\times_H X$ since this is like a tensor product in that $gh\otimes x=g\otimes hx$ for $h\in H$. Then $H\ltimes X$ is the groupoid with unit space $X$ and arrow space $H\times X$ where the arrow $(h,x)\colon x\to hx$. The product is $(h,h'x)(h',x)= (hh',x)$ and the topology is the product topology on $H\times X$ and the usual topology on $X$. The inverse is $(h,x)^{-1}=(h^{-1},hx)$. The groupoid $G\ltimes (G\times_H X)$ is defined similarly but has object space $G\times_H X$ and arrow space $G\times (G\times_H X)$.

To find a Morita equivalence I need a principal bibundle for these groupoids with the appropriate properties. The precise definition is on page 23 of the linked file.

Let $Z=G\times X$ with the product topology. We have open maps $p\colon Z\to G\times_H X$ and $q\colon Z\to X$ given by the quotient map in the first case and the projection in the second case. These can be use as anchors (or moment maps) for a left action of $G\ltimes (G\times_H X)$ and a right action of $H\ltimes X$ which commute. The left action of $G\ltimes (G\times H_X)$ is given by $(g_1,g_0\otimes x)(g_0,x) = (g_1g_0, x)$ and the right action of $H\ltimes X$ is given by $(g_0,x)(h,h^{-1}x) = (g_0h,h^{-1}x)$. It is easy to check that these are free and proper commuting actions. Also the quotient of $Z$ by the action of $G\ltimes (G\times_H X)$ is homeomorphic to $X$ via $q$ and the quotient of $Z$ by the action of $H\ltimes X$ is $G\times_H X$ by construction. Thus this bibundle gives a Morita equivalence of groupoids and hence a Morita equivalence of $C^*$-algebras.

Here is a proof that $C(X)\rtimes H$ is Morita equivalent to $C(X\times_H G)\rtimes G$. I'm assuming here that $G,H$ are countable discrete groups and $X$ is second countable. First of all, since I prefer to work with left actions, I will assume that $H$ acts on the left of $X$ and I will write instead $G\times_H X$ for $(G\times X)/H$ where the action of $H$ is on the right via $(g,x)h = (gh,h^{-1}x)$. I hope that is fine.

I will show that the corresponding transformation etale groupoids are Morita equivalent groupoids. See https://www.aidansims.com/files/GroupoidsNotes-2017.pdf section 3.4. It is a well-known result of Renault that Morita equivalent Hausdorff groupoids have strongly Morita equivalent $C^*$-algebras and the proof of Theorem 3.4.4 in the above reference tells you how to build the imprimitivity bimodule from the groupoid equivalence.

I will write $g\otimes x$ to denote the class of $(g,x)$ in $G\times_H X$ since this is like a tensor product in that $gh\otimes x=g\otimes hx$ for $h\in H$. Then $H\ltimes X$ is the groupoid with unit space $X$ and arrow space $H\times X$ where the arrow $(h,x)\colon x\to hx$. The product is $(h,h'x)(h',x)= (hh',x)$ and the topology is the product topology on $H\times X$ and the usual topology on $X$. The inverse is $(h,x)^{-1}=(h^{-1},hx)$. The groupoid $G\ltimes (G\times_H X)$ is defined similarly but has object space $G\times_H X$ and arrow space $G\times (G\times_H X)$.

To find a Morita equivalence I need a principal bibundle for these groupoids with the appropriate properties. The precise definition is on page 23 of the linked file.

Let $Z=G\times X$ with the product topology. We have open maps $p\colon Z\to G\times_H X$ and $q\colon Z\to X$ given by the quotient map in the first case and the projection in the second case. These can be use as anchors (or moment maps) for a left action of $G\ltimes (G\times_H X)$ and a right action of $H\ltimes X$ which commute. The left action of $G\ltimes (G\times H_X)$ is given by $(g_1,g_0\otimes x)(g_0,x) = (g_1g_0, x)$ and the right action of $H\ltimes X$ is given by $(g_0,x)(h,h^{-1}x) = (g_0h,h^{-1}x)$. It is easy to check that these are free and proper commuting actions. Also the quotient of $Z$ by the action of $G\ltimes (G\times_H X)$ is $X$ and the quotient of $Z$ by the action of $H\ltimes X$ is $G\times_H X$ by construction. Thus this bibundle gives a Morita equivalence of groupoids and hence a Morita equivalence of $C^*$-algebras.

Here is a proof that $C(X)\rtimes H$ is Morita equivalent to $C(X\times_H G)\rtimes G$. I'm assuming here that $G,H$ are countable discrete groups and $X$ is second countable. First of all, since I prefer to work with left actions, I will assume that $H$ acts on the left of $X$ and I will write instead $G\times_H X$ for $(G\times X)/H$ where the action of $H$ is on the right via $(g,x)h = (gh,h^{-1}x)$. I hope that is fine.

I will show that the corresponding transformation etale groupoids are Morita equivalent groupoids. See https://www.aidansims.com/files/GroupoidsNotes-2017.pdf section 3.4. It is a well-known result of Renault that Morita equivalent Hausdorff groupoids have strongly Morita equivalent $C^*$-algebras and the proof of Theorem 3.4.4 in the above reference tells you how to build the imprimitivity bimodule from the groupoid equivalence.

I will write $g\otimes x$ to denote the class of $(g,x)$ in $G\times_H X$ since this is like a tensor product in that $gh\otimes x=g\otimes hx$ for $h\in H$. Then $H\ltimes X$ is the groupoid with unit space $X$ and arrow space $H\times X$ where the arrow $(h,x)\colon x\to hx$. The product is $(h,h'x)(h',x)= (hh',x)$ and the topology is the product topology on $H\times X$ and the usual topology on $X$. The inverse is $(h,x)^{-1}=(h^{-1},hx)$. The groupoid $G\ltimes (G\times_H X)$ is defined similarly but has object space $G\times_H X$ and arrow space $G\times (G\times_H X)$.

To find a Morita equivalence I need a principal bibundle for these groupoids with the appropriate properties. The precise definition is on page 23 of the linked file.

Let $Z=G\times X$ with the product topology. We have open maps $p\colon Z\to G\times_H X$ and $q\colon Z\to X$ given by the quotient map in the first case and the projection in the second case. These can be use as anchors (or moment maps) for a left action of $G\ltimes (G\times_H X)$ and a right action of $H\ltimes X$ which commute. The left action of $G\ltimes (G\times H_X)$ is given by $(g_1,g_0\otimes x)(g_0,x) = (g_1g_0, x)$ and the right action of $H\ltimes X$ is given by $(g_0,x)(h,h^{-1}x) = (g_0h,h^{-1}x)$. It is easy to check that these are free and proper commuting actions. Also the quotient of $Z$ by the action of $G\ltimes (G\times_H X)$ is homeomorphic to $X$ via $q$ and the quotient of $Z$ by the action of $H\ltimes X$ is $G\times_H X$ by construction. Thus this bibundle gives a Morita equivalence of groupoids and hence a Morita equivalence of $C^*$-algebras.