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.



Torsion version of HNN extensions. 
It has been considered by Sergey V. Ivanov in "On HNN-extensions in the class of groups of large odd exponent". Groups, rings, Lie and Hopf algebras (St. John's, NF, 2001), 39–53, Math. Appl., 555, Kluwer Acad. Publ., Dordrecht, 2003. Particular cases of HNN extensions in the class of torsion groups were also used in our paper with Olshanskii, Non-amenable finitely presented torsion-by-cyclic groups. Publ. Math. Inst. Hautes Études Sci. No. 96 (2002), 43–169 (2003). 


