Let X be a compact connected Kähler manifold. We consider the category \(\mathcal {C}^\textrm{EC}(X)\) of flat holomorphic connections \((E,\, \nabla ^E)\) over X satisfying the condition that the underlying holomorphic vector bundle E admits a filtration of holomorphic subbundles preserved by the connection \(\nabla ^E\) such that the monodromy of the induced connection on each successive quotient has finite image. The category \(\mathcal {C}^\textrm{EC}(X)\) , equipped with the neutral fiber functor that sends any object \((E,\, \nabla ^E)\) to the fiber \(E_{x_0}\) , where \(x_0\, \in \, X\) is a fixed point, defines a neutral Tannakian category over \(\mathbb {C}\) . Let \(\varpi ^{\textrm{EC}}(X,\, x_0)\) denote the affine group scheme corresponding to this neutral Tannakian category \(\mathcal {C}^\textrm{EC}(X)\) . Let \(\pi ^{\textrm{EN}}(X,\, x_0)\) be an extension of the Nori fundamental group scheme over \(\mathbb {C}\) [8]. We show that \(\pi ^{\textrm{EN}}(X,\, x_0)\) is a closed subgroup scheme of \(\varpi ^{\textrm{EC}}(X,\, x_0)\) . Finally, we discuss an example illustrating that if X is not Kähler, then the natural homomorphism \(\pi ^{\textrm{EN}}(X,\, x_0)\, \longrightarrow \, \varpi ^{\textrm{EC}}(X,\, x_0)\) might fail to be an embedding.
