On HNN-extensions in the class of groups of large odd exponent
  A sufficient condition for the existence of HNN-extensions in the class of groups of odd exponent $n \gg 1$ is given in the following form. Let $Q$ be a group of odd exponent $n > 2^{48}$ and $\mathcal G$ be an HNN-extension of $Q$. If $A \in \mathcal G$ then let $\mathcal F(A)$ denote the maximal subgroup of $Q$ which is normalized by $A$. By $\tau_A$ denote the automorphism of $\mathcal F(A)$ which is induced by conjugation by $A$. Suppose that for every $A \in \mathcal G$, which is not conjugate to an element of $Q$, the group $<\tau_A, \mathcal F(A)>$ has exponent $n$ and, in addition, equalities $A^{-k} q_0 A^{k} = q_k$, where $q_k \in Q$ and $k =0, 1, ..., [2^{-16}n]$ ($[2^{-16}n]$ is the integer part of $2^{-16}n$), imply that $q_0 \in \mathcal F(A)$. Then the group $Q$ naturally embeds in the quotient $\mathcal G / \mathcal G^n$, that is, there exists an analog of the HNN-extension $\mathcal G$ of $Q$ in the class of groups of exponent $n$. 
S.V. Ivanov