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 particular, 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 $G \rtimes A$.



Irreducible reps and characters of $G \rtimes A$
I might as well turn my comment into an answer. I will just write $GA$ for the semidirect product ( with the normal subgroup being $G$). Clifford's theorem outlines a procedure for computing the character table, but it is usually not straightforward in practice.


Firstly, we need to compute a set of orbit representatives for the complex irreducible characters of $G$ under the action of $A$.


Then, if the irreducible character $\chi$ of $G$ (one orbit representative) has stabilizer $A\_{\chi}$, we need to find all irreducible characters of $GA\_{\chi}$ whose restriction to $G$ is a multiple of $\chi$. In practice, this is usually the most difficult step, and often involves working with central extensions of $A\_{\chi}$ and the determination of $2$-cocycles.


Once the irreducible characters of the required type for $GA\_{\chi}$ have been determined, we induce them all from $GA\_{\chi}$ to $GA$.


Doing this for each orbit gives all irreducible characters of $G$ exactly once.


When $A$ is cyclic, this procedure simplifies quite a bit, and even more so when $A$ has prime order.


For example, in the case you are looking at, of $G \langle t \rangle$ with $t$ of order $2$, it is necessary to determine the orbits of $t$ on the irreducible characters of $G = M\_{12}$.


Once these are found, if the irreducible character $\chi$ of $M\_{12}$ lies in an orbit of length $2$, then ${\rm Ind}\_{G}^{G\langle t \rangle}(\chi)$ is an irreducible character of $G\langle t \rangle$ which vanishes identically outside $G$ and agrees with $\chi + \chi^{t}$ on $G$.


However, if an irreducible character $\chi$ lies in an orbit of length $1$, so is $\langle t \rangle$ invariant, there are exactly two irreducible extensions of $\chi$ to $G \langle t \rangle$, say ${\tilde \chi}$ and $\lambda {\tilde \chi}$, which differ by multiplication by the unique linear character $\lambda$ of $G\langle t \rangle$ with kernel $G$ (so that $\lambda(g) = -1$ for all $g \in G \backslash \langle t \rangle$).


There is work to do in determining the values of ${\tilde \chi}$ outside $G$.


(Of course, an alternative strategy in this case is to look up the character table in the Atlas!).


Later edit: I see that Derek Holt's comment, made while I was writing, has indicated the same procedures in the special case under consideration.


