Mackey theory for semidirect products: equivalence between constructions for modules I am learning about the so-called "Mackey Machine" for unitary irreps of semidirect products of locally compact groups. Let $G = N \rtimes H$ where $N$ is a closed normal abelian subgroup and $H$ is a closed subgroup which acts on $N$ by $\phi: H \to Aut(N)$. Given an unitary irrep $\sigma$ of $H$ and a irreducible representation $p$ of $N$, I have found two (I believe equivalent) ways in which an irrep of $G$ is constructed by different authors.
If $H_p$ is the semigroup of $H$ which stabilizes $p$, and $G_p = N \rtimes H_p$, then we have the following commutative diagram of inclusions.
$\require{AMScd}$
\begin{CD}
H_p @>>> H\\
@VVV @VVV\\
G_p @>>> G
\end{CD}
If I am understanding it correctly, Folland, in A Course in Abstract Harmonic Analysis (see the discussion leading to Theorem 6.43, on pp. 199-201 of the second edition) goes "first down then right": defines a representation of $G_p$ as $p\otimes \sigma$, then takes the representation of $G$ induced by it (i.e. the same way it is presented in this question). On the other hand, Etingof et al. - Introduction to representation theory (section 4.26 on pp. 76) takes the route "right then down": first they consider the representation of $H$ induced by $\sigma$, then they extend it to $G$ with a skew-product.
Question 1: How to express these two constructions in terms of modules?
Question 2: Since the two constructions are equivalent, the modules resulting from the first and the second path should be isomorphic. Can we see that isomorphism explicitly?
Let me pretend that the groups are finite, since I believe it should work equally well without any topological complicacies.
What I tried
Let $\mathcal{H}_\sigma$ the $\mathbb{C}(H_p)$-module given by $\sigma$. Since $p$ is a 1-dimensional representation of $N$, we can give to $\mathbb{C}$ the structure of a $\mathbb{C}(N)$ module, which I will denote with $\mathbb{C}_p$. Then Folland's construction is (I am keeping the structure of the diagram above, to match each module to the corresponding group, but the arrow have no further meaning):
\begin{CD}
\mathcal{H}_\sigma \\
@VVV \\
\mathbb{C}_p \otimes_{\mathbb{C}(H_p)} \mathcal{H}_\sigma @>>> \mathbb{C}(G) \otimes_{\mathbb{C}(G_p)} \mathbb{C}_p \otimes_{\mathbb{C}(H_p)} \mathcal{H}_\sigma
\end{CD}
Let me explain what I think should be going on: $\mathbb{C}_p$ can be made into a $(\mathbb{C}(G_p), \mathbb{C}(H_p))$-bimodule, thanks to the action $\phi$ (which stabilizes $p$).
But I am not sure what module is given by the other route
\begin{CD}
\mathcal{H}_\sigma @>>> \mathbb{C}(H) \otimes_{\mathbb{C}(H_p)} \mathcal{H}_\sigma \\
@. @VVV\\
@. ?
\end{CD}
 A: By request, I post my comments (1 2).  I think that they answer the revised question, but let me know if not.
Question 1.  Folland's construction corresponds to extending a $\mathbb C(H_p)$-module $\mathcal H_\sigma$ of $H_p$ across $N$ by $p$ to $G_p = N \rtimes H_p$, giving $\mathbb C_p \otimes_{\mathbb C} \mathcal H_\sigma$; and then inducing up to $G$, giving $\mathbb C(G) \otimes_{\mathbb C(G_p)} (\mathbb C_p \otimes_{\mathbb C} \mathcal H_\sigma)$.  (I'm not sure how to make sense of your proposed $\mathbb C_p \otimes_{\mathbb C(H_p)} \mathcal H_\sigma$, since $\mathbb C_p$ is not a $\mathbb C(H_p)$-module in any obvious-to-me way; notice that, since $\mathbb C_p$ is $1$-dimensional, specifying the module structure would be equivalent to specifying a homomorphism $H_p \to \mathbb C^\times$.  The tensor product over $\mathbb C(H_p)$ will collapse to a $1$-dimensional representation, on which $H_p$ acts trivially (if $\mathcal H_\sigma$ is not isomorphic to $\mathbb C_p$ as $\mathbb C(H_p)$-modules) or by the relevant character (if they are isomorphic).)
Etingof et al.'s construction corresponds to inducing $\mathcal H_\sigma$ up to $H$, giving $\mathbb C(H) \otimes_{\mathbb C(H_p)} \mathcal H_\sigma$; and then extending across $N$ by $p$ to $G = N \rtimes H$, giving $\mathbb C_p \otimes_{\mathbb C} (\mathbb C(H) \otimes_{\mathbb C(H_p)} \mathcal H_\sigma)$.
Question 2.  The isomorphism from Folland's to Etingof's construction is given by $(n \rtimes h) \otimes_{\mathbb C(G_p)} (z \otimes_{\mathbb C} v) \mapsto p(n)z \otimes_{\mathbb C} (h \otimes_{\mathbb C(H_p)} v)$.  The inverse isomorphism is given by $z \otimes_{\mathbb C} (h \otimes_{\mathbb C(H_p)} v) \mapsto (1 \rtimes h) \otimes_{\mathbb C(G_p)} (z \otimes_{\mathbb C} v)$.
