Suppose $N \lhd G$. Given isomorphism $i: N \rtimes_\pi G/N \to G$ and action $\pi: G/N \times N \to N$, show how to obtain a splitting $\varphi: G/N \to G$ of the quotient map $G \to G/N$.
Here, we call $\varphi$ a "splitting" of the quotient homomorphism $G\to G/N$ if it is a homomorphism $\varphi: G/N\to G$ such that $\varphi(gN)\in gN$ for every $gN\in G/N$.
I think $\varphi$ given by $\varphi(gN) = i(e, gN)$ should be the desired splitting. I've shown that it is a homomorphism, but am stuck with proving that it satisfies $\varphi(gN) \in gN$ for every coset $gN$.
So, in particular, how can we show that $\varphi(gN) = i(e, gN) \in gN?$