Let $G$ be a topological group. Denote the same group with the discrete topology by $G^\delta$ and denote the group of connected components of $G$ by $\pi_0G$. I am interested in the question when we can find a section to the canonical map $G \to \pi_0G$.

An obstruction to do that is the requirement that the map $$H^{\ast}(BG^{\delta}) \leftarrow H^{\ast}(B\pi_0G): \alpha$$ (cohomology with $\mathbb Z$ coefficients) induced by $G^{\delta} \to \pi_0G$ has to be split injective.

(side remark: This is exactly how one proves Morita's theorem: For $g \geq 5$, the mapping class group of a genus $g$ surface can not be realized by diffeomorphisms: Here $G = \text{Diffeo}(S_g)$, $\pi_0G = \text{Mod}_g$ and the class $\kappa_3 \in H^6(BG) = H^6(B\text{Mod}_g)$ lies in the kernel of the map $\alpha$ mentioned above. A reference for this is Morita's book on characteristic classes of surface bundles, page 172.)

My question is about the case where this obstruction vanishes: Does someone know an example of a topological group $G$ where the discussed map $\alpha$ is split injective, but still the map $G \to \pi_0 G$ does not admit a section?