Let $G$ be central extension of an abelian group $A$ by some group $H$.
Is it possible to characterize all irreducible representions of $G$ 
in terms of irreducible representations of $A$ and $H$?



Representations of central extensions
The question is somewhat unprecise. I assume you mean irreps over $\mathbb{C}$ and *finite* groups. Then the answer is "no" as long as you aren't more specific about the central extension in question. For example, four of the five nonisomorphic groups of order $p^3$ can be obtained as central extensions of $C\_p$ by $C\_p\times C\_p$, and the irreps of the abelian and the nonabelian ones are quite different. (By the way, it seems more usual to call $G$ a central extension of $H$ by $A$, assuming you want to have $A\subseteq \operatorname{\textbf{Z}}(G)$.)  

However, the *projective representations* of $H$ tell you something about representations of central extensions, and here Schur's theory of the *Schur multiplier* and the "*Darstellungsgruppe*" of $H$ is helpful. You should look for these keywords in books on representation theory (e. g., the appropriate sections in Huppert, *Endliche Gruppen I*, Kapitel V, contain much information).


