I have a HNN extension $\displaystyle G=<t,A;t^{-1}ht=h>$ where $\displaystyle A=L<h>$ is a semi-direct product of $\displaystyle L$ by $\displaystyle <h>$ and $\displaystyle <h>$ is cyclic.

$\displaystyle G$ can be written as free products of $\displaystyle A$ and $\displaystyle B$ amalgamating $\displaystyle <h>$ where $\displaystyle B=<t,h;t^{-1}ht=h>$ is a free abelian group.

Can $\displaystyle B$ expressed as a semi-direct product of cyclic free group $\displaystyle <t;->$ by $\displaystyle <h>$?