Exercise 7.6.2.5. Let $G$ be a group and let $H \subseteq G$ be a commutative normal subgroup, so that we have a commutative diagram of Kan complexes
Show that (7.63) is a pullback diagram in the ordinary category of Kan complexes, and that it determines a pullback diagram in the $\infty $-category $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan})$ (see Example 7.6.3.2).
Show that, if $H$ is contained in the center of $G$, then the diagram (7.63) is also pullback square in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.
Show that, if $H$ is not contained in the center of $G$, then the diagram $B_{\bullet } G \rightarrow B_{\bullet } (G/H) \leftarrow \Delta ^0$ does not have a limit in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. In particular, the diagram (7.63) is not a pullback square in $\mathrm{h} \mathit{\operatorname{Kan}}$.