(Note that I've edited the main body of the question to make it clear for other readers.)

Fix a principal $G$-bundle $\rho: P \rightarrow X$ and fix a point $p \in P_x$ in the fiber above $x \in X$. If $\rho$ is equipped with a flat connection $\omega$, we get a surjective homomorphism $\pi_1(X,x) \rightarrow \text{Hol}_p(\omega)$.

- When is the map $\pi_1(X, x) \rightarrow \text{Hol}_p(\omega) \hookrightarrow G \rightarrow \text{Inn}(G)$ surjective?

The motivation for this question comes from the fact that such a composition is surjective in the case of ``parallel transport'' of fundamental groups in $X$ along curves changing basepoints.