Let $G$ be a group and $A$ a trivial $G$-module. The Universal Coefficient Theorem yields a split exact sequence
$$0\longrightarrow {\rm Ext}(H_{n-1}(G),A)\longrightarrow H^n(G,A)\longrightarrow {\rm Hom}(H_n(G),A)\longrightarrow 0.$$
In terms of cohomology classes, what is the explicit description of the map ${\rm Ext}(H_{n-1}(G),A)\to H^n(G,A)$?
In the case $n=2$, the map is induced by the epimorphism $G\to G^{\rm ab}$.
Additional questions: How does the splitting map ${\rm Hom}(H_n(G),A)\to H^n(G,A)$ act, again explicitly in terms of homomorphisms $H_n(G)\to A$ and elements of $H^n(G,A)$? Or maybe, if it's easier, what is the splitting map $H^n(G,A)\to {\rm Ext}(H_{n-1}(G),A)$?
Edit: As the first comment below shows, splitting maps may not have nice explicit descriptions.