I am familiar with the definition of the HNN extension of a group relative to an isomorphism between two of its subgroups. For comparison's sake let me make that explicit. For groups $G_1, G_2\leq G=\langle S\mid R\rangle$ with isomorphism $\alpha:G_1\rightarrow G_2$, and new element $t\notin S$, we denote $G*_{\alpha}:=\langle S,t\mid R, \forall g\in G_1 \;tgt^{-1}=\alpha(g)\rangle$.
What I want to know is, what is meant by the notation $G*_H$, where $H$ is just some other group? In case the following is needed to make sense of this, in what I'm reading all my groups are finitely generated, and Kleinian, and the group $H$ is non-elementary (though I expect the notation has a definition in a broader setting).
Does the notation imply that there are isomorphisms from $H$ to two possibly different subgroups of $G$, implying existence of a $G_1, G_2$, and isomorphism $\alpha$ between them as in the more familiar notation described in the 1st paragraph? If so, doesn't the structure of $G*_H$ depend a lot on the choices for these objects? How can we get a well defined extension when we only specify $H$?