Let $1\rightarrow G{\rightarrow}^\varphi E\rightarrow^\psi H\rightarrow 1$ be short exact sequence of groups and homomorphisms where $E$ is a semidirect product $E=G \rtimes H$.
I am struggling to see why does this short exact sequence split? That is, why is there a homomorphism $\gamma : H\rightarrow E$, such that $\psi \gamma = 1_H$?
edit: fixed $1_H$.