I'm interested in cohomology operations (in ordinary cohomology)
$$H^i(-, G)\rightarrow H^{i+1}(-, H)\;,$$
that is, elements of
$$H^{i+1}(K(G, i), H)\;.$$
I know that $K(G, 1)=BG$, so for $i=1$, those cohomology operations are in $H^2(BG, H)$, and therefore given by the Bocksteins of the corresponding central extensions of $G$ by $H$. Also, for $G=H=\mathbb{Z}_p$, the **stable** cohomology operations are given by the Steenrod algebra, and the only degree-1 elements are Bocksteins.

However, I don't know how it is for unstable cohomology operations, or groups other than $\mathbb{Z}_p$. I'm mostly interested in simple groups, such as finitely generated or $\mathbb{R}/\mathbb{Z}$.