In order to advance my research I'm supposed to understand this fact:

Let $A$ be an abelian group and $S$ a finite group acting on $A$. This defines the semi-direct product $A\rtimes S$:

Let $\chi$ be a character of $A$. Define $S_\chi = \{s\in S \mid s\chi = \chi\}$ (Recall: $s\chi = \chi \iff \forall a\in A, \chi(sas^{-1}) = \chi(a)$ ).

Note this defines a subgroup of $A \rtimes S> A \rtimes S_\chi$

We can extend each representation of $A$ to a representation of $A \rtimes S_\chi$ by: $\chi: A \rightarrow \mathbb{C}, \chi(as) := \chi(a)$.

Let $(\sigma,V) \in Irr (S_\chi)$. Note $\chi \otimes \sigma $ is a representation of $A \rtimes S_\chi$. Then:

1. $Ind_{A \rtimes S_\chi}^{A \rtimes S} \chi \otimes \sigma$ is irreducible

2. For any irreducible representation of $A \rtimes S$, there are $\chi \in Irr(A)$ and $\sigma \in Irr(S_{\chi})$ s.t. the representation is: $Ind_{A \rtimes S_\chi}^{A \rtimes S} \chi \otimes \sigma$

If I understand correctly the proof of this statement should be straight forward from Mackey theory, but I can't see the picture. My instructor calls this statement Mackey theory - which is different from the statement I see in other textbooks (seems like this is an application - correct me if I'm wrong).