Flat Maurer-Cartan connection iff flat Berry connection

Question

I am studying two connections on induced representation spaces $\text{Ind}_{H}^{G} \Gamma$, where $H \subseteq G$ are groups, and $\Gamma$ is an irrep of $H$.

1. The first is the canonical or $H$-connection, which is the projection of $G$'s Maurer-Cartan form into $H$'s Lie algebra. A paper by Milnor classifies all such flat connections in terms of monodromy groups $\pi_1(G/H)/\ker \rho$, induced by $\pi_1$-irreps $\rho$. These irreps determine the monodromy after a non-contractible closed path in the coset space.

2. The second is the Berry connection (see here and here for mathematican description), $\langle a,\mu |\partial a,\nu\rangle$, for "position states" $|a,\mu\rangle$ of a particle on the induced-rep space, where $a \in G/H$, and $\mu,\nu$ index the irrep space of $\Gamma$. Here a position state moves in the coset space $G/H$, and obtains a holonomy (in physics: Berry phase or matrix) after a closed path that acts on the $\mu$ factor. The possible monodromies (physics: topological Berry phases or matrices) for non-contractible closed paths are classified by the monodromy group $H/\ker \Gamma$.

The two connections are related but not generally equal [Eq. (4.2) here or here]. For all examples I'm aware of, the connections are zero iff the two monodromy groups match. In other words, I see that both connections are flat for a given inducing $H$-irrep $\Gamma$ if there exists a $\pi_1$-irrep $\rho$ such that

$$ \frac{H}{\ker \Gamma} = \frac{\pi_1(G/H)}{\ker \rho}~. $$

I was hoping someone could either prove this or provide a counterexample. Here are some examples:

• Both connections are flat and the monodromy is trivial for the sphere, $\text{Ind}_{U(1)}^{SU(2)} 0$, where $0$ is the trivial irrep of $U(1)$. The above equation holds since $H/\ker\Gamma = U(1)/U(1)$, $\pi_1$ is trivial, and its only (trivial) irrep can be picked to be $\rho$.

• Both connections are not flat and the monodromy is trivial for the sphere with a monopole of nonzero strength $\lambda$, $\text{Ind}_{U(1)}^{SU(2)} \lambda$, where $\lambda$ is a non-trivial irrep of $U(1)$. Conjecture holds since $H/\ker \Gamma = U(1)/\mathbb{Z}_{|\lambda|} = U(1)$, $\pi_1$ is the same (i.e., trivial), and so it's impossible to find a $\rho$ that can fill in the above equation.

• Both connections are flat and the monodromy is nontrivial for the Poincare homology sphere with some inducing $A_5$-irrep $\Gamma$, $\text{Ind}^{SO(3)}_{A_5} \Gamma$. The above equation holds since $\pi_1 = 2A_5$, and so one can pick the $\pi_1$-irrep corresponding to $\Gamma$ under the double cover to be $\rho$.