I have the following definition of a split extension:
An extension $G$ of $H$ by $N$ is called a split extension if the canonical projection $\pi:G \to G/N$ ($G/N$ is isomorphic with $H$) has a section $\sigma$ (i.e.: an injection from $H$ to $G$ with $\pi\circ\sigma=1$).
But now I'm confused because I thought this is always true (by the axiom of choice). Can someone explain me please?
In the context of groups, you usually want to have all your maps to be structure preserving. Your definition of section of a group homomorphism $\pi$ as an injection $\sigma$ such that $\pi \circ \sigma = id$ thus needs to be amended: We want $\sigma$ not just to be any injection but to be an injective homomorphism.
You are right that the axiom of choice gives you for any surjection an injection which is a right inverse to it, but this is how things work in the context of sets and arbitrary maps. In the context of groups and group homomorphisms, things are different.
For example, take $G = \mathbb{Z}$ and $N = 2 \mathbb{Z}$, then $\pi$ is the morphism $\mathbb{Z} \mapsto \mathbb{Z}/2\mathbb{Z}$. Note that there exists no morphism $\mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}$ of groups so, in particular, $\pi$ has no right inverse and thus we have an example of a non-split extension.