I am thinking of a version of HNN extensions as follows:

Assume $H,K$ are subgroups of a group $G$ and $\phi:H\to K$ is an isomorphism. We define $ G_{\phi,n}$ to be the group generated by $G$ and $x\notin G$ satisfying the conditions $x h x^{-1}=\phi(h)$ and $x^n=1$.

I was wondering if such a construction has appeared in the literature before? What are the main properties (and the name) of these groups?

I appreciate any references.