Is every non-simple finite group constructible from simple groups of lesser order by extensions?

Question

The question is easy to formulate: Let be $G$ a finite group with some normal subgroup, let's say $N \triangleleft G$ where $N\neq \{1\}$ and $N\neq G$. The question is, there exists an group homomorphism $\alpha : N\hookrightarrow G$ and other group homomorfism $\beta : G \twoheadrightarrow G/N$ such $im \; \alpha \; = \; ker\; \beta$? As in the commentaries say me, it's trivial $\alpha\equiv Id_N$ and $\beta\equiv \pi: G \rightarrow G/N :: x \mapsto xN$, is a first immediate answer. Then, the question is, can every non-simple finite group be constructible from simple groups of lesser order by extensions?