Second cohomology of a rank-two free abelian group with truncated group-ring coefficients
= Second cohomology of a rank-two free abelian group with truncated group-ring coefficients
If $G\cong\mathbb Z^2$, $I$ is the augmentation ideal, and $M=\mathbb ZG/I^2$, then
$$
H^2(G,M)\cong M/IM\cong\mathbb Z.
$$
The quotient map $M\to\mathbb ZG/I\cong\mathbb Z$ induces an isomorphism in second cohomology.