Let $A$ be a group. Then we choose that $B$ is a subgroup of $A$.
Let us write the cohomology group cocycle of $A$, as $\alpha_{d}(\{a\}) \in H^d(A,U(1))$ where $\{a\}$ is a shorthand for a set of $a_j \in A$ for $j=1,2, \dots$.
(1) For any given subgroup $B$ (here $B$ may not be a trivial group), what are the conditions on the subgroup $B \subset A$ that the injective group homomorphism $$B \overset{i}{\to} A$$ such that $$ \alpha_{d}(\{b\})$$ is a coboundary (namely, a trivial element) in $H^d(B,U(1))$?

(2) What are the conditions that there exists a group homomorphism $$A \overset{f}{\to} B$$ where we name $f(a)=b$ for any $a \in A$ and some $b \in B$, such that $$\alpha_{d}(\{f(a)\})= \alpha_{d}(\{b\})$$ is a coboundary (namely, a trivial element) in $H^d(B,U(1))$?
We can consider $A$ and $B$ are finite groups.
p.s. Bonus questions: What if $A$ and $B$ are Lie groups?