Let $G$ be a group (you preferred type: finite, compact, ...):

Mackey has a machinery to classify all irreducible representations of a locally compact group $G$ in terms surjective group homomorphism: $$ \sigma : G \rightarrow N$$ by some irreducible representations of subgroups of $N$ and some projective representations $G/ kern(\sigma)$, where $kern(\sigma)$ is a so called type-1 subgroup (i.e. the von Neumann group algebra is a direct integral type $1$ factors), e.g. take a finite, abelian, compact or amenable group.

Q1: Is there a nice reference for a condensed treatment of these results?

Consider an exact extension of $G$ Hilbert modules: $$ A \rightarrow B \rightarrow C.$$

Q2: When and how can we classify the decomposition the irreducible subrepresentation of $B$ in terms of data coming $A$ and $C$?

Everthing in the case of finite groups would already be satisfying for me.

Let $G$ be a group (you preferred type: finite, compact, ...):

Mackey has a machinery to classify all irreducible representations of a locally compact group $G$ in terms surjective group homomorphism: $$ \sigma : G \rightarrow N$$ by some irreducible representations of subgroups of $N$ and some projective representations $G/ kern(\sigma)$, where $kern(\sigma)$ is a so called type-1 subgroup (i.e. the von Neumann group algebra is a direct integral type $1$ factors), e.g. take a finite, abelian, compact or amenable group.

Q1: Is there a nice reference for a condensed treatment of these results?

Let $G$ be a group (you preferred type: finite, compact, ...):

Mackey has a machinery to classify all irreducible representations of a locally compact group $G$ in terms surjective group homomorphism: $$ \sigma : G \rightarrow N$$ by some irreducible representations of subgroups of $N$ and some projective representations $G/ kern(\sigma)$, where $kern(\sigma)$ is a so called type-1 subgroup (i.e. the von Neumann group algebra is a direct integral type $1$ factors), e.g. take a finite, abelian, compact or amenable group.

Q1: Is there a nice reference for a condensed treatment of these results?

Consider an exact extension of $G$ modules: $$ A \rightarrow B \rightarrow C.$$

Q2: When and how can we classify the decomposition the irreducible subrepresentation of $B$ in terms of data coming $A$ and $C$?

Everthing in the case of finite groups would already be satisfying for me.

Let $G$ be a group (you preferred type: finite, compact, ...):

Mackey has a machinery to classify all irreducible representations of a locally compact group $G$ in terms surjective group homomorphism: $$ \sigma : G \rightarrow N$$ by some irreducible representations of subgroups of $N$ and some projective representations $G/ kern(\sigma)$, where $kern(\sigma)$ is a so called type-1 subgroup (i.e. the von Neumann group algebra is a direct integral type $1$ factors), e.g. take a finite, abelian, compact or amenable group.

Q1: Is there a nice reference for a condensed treatment of these results?

Consider an exact extension of $G$ Hilbert modules: $$ A \rightarrow B \rightarrow C.$$

Q2: When and how can we classify the decomposition the irreducible subrepresentation of $B$ in terms of data coming $A$ and $C$?

Everthing in the case of finite groups would already be satisfying for me.