Let $G, G_1, G_2$ be compact Lie groups with homomorphisms $f_1:G_1 \to G$ and $f_2: G_2\to G$. Let $P_1, P_2$ be principal bundles for $G_1,G_2$ and assume that the bundles $P_i\times_{G_i} G$ are both isomorphic (by fixed isomorphisms) to a bundle $P$.

Let now $H$ be the pullback of the group diagram given by $f_1$ and $f_2$. Let $Q$ be the (topological) pullback of the induced diagram given by $P_1$, $P_2$ and $P$.

Now $Q$ has "fibres" equivalent to $H$, but does $Q$ always form a (locally trivial) principal $H$-bundle?