Is there a theorem which classifies irreducible representations of semi-direct product of finite groups $G \rtimes A$, where $A$ is a finite abelian group and hence write down the character table for $G \rtimes A$? In particle, I want to write down the character table for $M_{12} \rtimes \mathbb{Z}_2$ from the character table of $M_{12}$.

Serre's book on representation theory has a theorem on groups of the form $A \rtimes G$ for $A$ abelian, but I am studying groups of the form $G \rtimes A$.