I would like some help on the following problem from anyone who would like to help.
Let $f: H \to G$ be a group homomorphism. For $h \in H$, define $\rho(h) = \phi_{f(h)} \in Aut(G)$.
The situation being as described, prove that the semi-direct product $G\rtimes_{\rho} H$ is isomorphic to the direct product $G \times H$.
Help will be greatly appreciated!