I have a question.
Let be a HNN extension of base group with stable letter .
Is it true that ?
I think is it true.
As an HNN extension , we have that in fact , for some isomorphism .
Since you give , this means you chose = the inverse involution automorphism of , so your group is the amalgamated product (as any other HNN extension) of with itself via the above involution.
As t is a "foreign" letter to the group then clearly
My point is that the result is clear and is the whole point of HNN-extensions! Although it does still require proof...