I am currently trying to solve an exercise that consists of two parts:
Let $H$ be a group and $A \leq H$ be a subgroup such that there is an injection $\phi: A \rightarrow H$. Let $G$ be the HNN-extension of $H$ by $\phi$. Let further $F = \langle \star_{i \in \mathbb{Z}} H_i | a_{i+1} = \phi(a)_i \rangle $ such that every $H_i$ is a copy of $H$ and $h \mapsto h_i$ identifies $H$ with $H_i \subseteq F$. Show that $G$ is isomorphic to a semidirect product of $F$ and $\mathbb{Z}$.
Show that the HNN extension obtained through the isomorphism $\mathbb{Z} \rightarrow 3\mathbb{Z}$ can be described as a semidirect product of $\mathbb{Z}$ and a subgroup of $\mathbb{Q}$.
I have no idea for the first case and in the second case, I see that the presentation of the HNN extension should be $\langle a, t | t^{-1}at = a^3\rangle$ and this looks very similar to the presentation of a semidirect product, but I don't find the right homomorphism and the subgroup of $\mathbb{Q}$.