Inspired by this post, I wonder, for a short exact sequence for some finite groups $A,B,C$, (I am happy to consider Lie groups too) $$1\to A \to B \to C \to 1,$$ whether there are generic group homomorphism for their cohomology groups?
$$H^d(B,M) \to H^d(A,M)?$$
$$H^d(B,M) \to H^d(C,M)?$$
$$H^d(C,M) \to H^d(B,M)?$$
$$H^d(A,M) \to H^d(B,M)?$$
For any $d$ or certain $d$?
Here $M$ can be $\mathbb{R}/\mathbb{Z}$ or $\mathbb{Z}$.