Let $A$ be a $G$-module where $G$ is a finite group. Let $B= A \oplus \mathbb{Z}[G]$. $H^0(H,B)$ is assumed to be cyclic of order $|H| \ \forall H \leq G$. Assume that $a_0+N_GA$ is the generator of $H^0(G,A)$. Consider the map $f:\mathbb{Z} \to B$, where $f(n)=na_0+nN_G$ where $N_G= \sum _{\sigma \in G} \sigma$. The map is injective. Prove that the induced cohomological map $\tilde{f}: H^0(H,\mathbb{Z}) \to H^0(H,B)$ is an isomorphism.
This appears in a proof of Tate's theorem in Part I, Chapter 7 in Class Field Theory -The Bonn Lectures by Jurgen Neukirch.
We could show either injectivity or surjectivity as both cohomology groups are cyclic of the same order.
I have attempted two proofs in the comments. The first one is incorrect. I believe that the second one is correct. If there is another approach, please let me know