Let's say that we have two groups $ G $ and $ H $ with an epimorphism $ \phi : G \to H $. Denote $ B $ to be the kernel of $ \phi $. If we have another group $ C $ isomorphic to $ B $ and a homomorphism $ \psi : H \to Aut(C) $, does it imply that $ G $ is isomorphic to the semidirect product of $ C $ and $ H $?
In the case of direct products, I guess we can say that $ G $ is isomorphic to the direct product of $ B $ and $ H $, but this other homomorphism $ \psi $ confuses me in a way that it feels like the isomorphism must also preserve something about $ \psi $.