I apologize in advance if this question is inappropriate for this forum.

I am reading Hatcher's book and have the following problem: Let $X$ be a space, $G$ an abelian group and $R$ a ring. In his proof of the universal coefficient theorem in $\S$ 3.1, he considers a natural homomorphism $h\colon H^k(X;G)\to \text{Hom}(H_k(X),G)$. However, when computing cup product structures in $\S$ 3.2, specifically in examples 3.8 and 3.9, he seems to be using a homomorphism $\tilde{h}\colon H^k(X;R)\to \text{Hom}(H_k(X;R),R)$.

I think I can see that such a homomorphism always exists when $R$ is a ring. More precisely, given a cocyle $\phi\colon C_k(X)\to R$, we can define $\tilde{\phi}\colon Z_k(X)\otimes R\to R$ by $\tilde{\phi}(\alpha \otimes r)=\phi(\alpha)r$, and the fact that $\delta\phi=\phi\partial=0$ means that this gives rise to a homomorphism $H_k(X;R)\to R$. In other words, the construction yields a map $\tilde{h}\colon H^k(X;R)\to \text{Hom}(H_k(X;R),R)$.

What I would like to know is:

(a) Is there a way to define something similar to $\tilde{h}$ with $G$ in place of $R$?

(b) Is this map $\tilde{h}$ surjective as $h$ is?

(c) What is its kernel?

Thank you.