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, goes "first down then left": 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, this paper 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.

I am trying to understand both constructions in terms of modules: the fact that the two constructions are equivalent should give a module isomorphism. Let me pretend that the groups are finite, since I believe it should work equally well without any topological complicacies.

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

\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 $p$).

The other route is a bit more involved \begin{CD} \mathcal{H}_\sigma @>>> \mathbb{C}(H) \otimes_{\mathbb{C}(H_p)} \mathcal{H}_\sigma \\ @. @VVV\\ @. \mathbb{C}_p \rtimes_\phi \mathbb{C}(H) \otimes_{\mathbb{C}(H_p)} \mathcal{H}_\sigma \end{CD} where the skew product $\rtimes_\phi$ is defined by the multiplication $(a, h)(b,k)=(a\phi_h(b), hk)$.

From this I am led to wonder

Question: is $\mathbb{C}_p \rtimes_\phi \mathbb{C}(H)$ a $(\mathbb{C}(G), \mathbb{C}(H_p))$-bimodule which is isomorphic to $\mathbb{C}(G)\otimes_{\mathbb{C}(G_p)} \mathbb{C}_p$?