Let $G$ be a group with a subgroup $A$, and let $\varphi:A\rightarrow G$ be any injection. The HNN extension with base $G$ and associated subgroups $A$ and $\varphi(A)$ is defined as $$G^{\ast}=\langle\; S_{G},\; t\;|\;R_{G},\;t^{-1}at=\varphi(a),\;a\in A\;\rangle,$$ where $\langle\; S_{G}\;|\;R_{G}\;\rangle$ is a presentation for $G$.
Is there any method or theorem that one could use to compute the (co)homology of $G^{\ast}$?
Let $\varphi,\psi: A\to G$ be two embeddings of a group $A$ to a group $G$ and denote by $G^*$ the corresponding HNN-extension. Robert Bieri in his article proved that for any $G^*$-module $M$ there are long exact sequences for homology $$ \to H_n(A,M) \to H_n(G,M) \to H_n(G^*,M) \to H_{n-1}(A,M) \to $$ and cohomology $$ \to H^n(G^*,M) \to H^n(G,M) \to H^n(A,M) \to H^{n+1}(G^*,M) \to $$