Studying a course on geometry and groups, I fell on the following property (which is not given as an exercise, but rather as an observation).
Let $A$ be any group, let $B \cong \langle t \rangle \cong \mathbb{Z}$ and let $\varphi_t \in Aut(A)$. Then the semi-direct product $A \rtimes_{\varphi} \mathbb{Z}$ is isomorphic to a HNN extension $A*_A$.
There are no more precision nor details. I don't really see how to prove this. Does anyone know how to?
Thanks in advance.