at.algebraic topology – The notion of a “relatively” flat connection

Suppose that $$X$$ is a connected smooth manifold and $$Gamma$$ is a group acting smoothly, freely, properly and discretely on $$X$$, so that $$Y=X/Gamma$$ is another smooth manifold endowed with a covering map $$pi:Xrightarrow Y$$.

Suppose that $$G$$ is a Lie group and that $$rho:Gamma rightarrow G$$ is a group homomorphism. Then we can consider the quotient $$E_rho =(Xtimes G)/Gamma$$, where the action of $$Gamma$$ on $$G$$ is given by composing $$rho$$ with the adjoint action of $$G$$ on itself. $$E_rho$$ is naturally a principal $$G$$-bundle over $$Y$$.

My question is if there exists a condition for a principal $$G$$-bundle $$E$$ on $$Y$$ to be isomorphic to an $$E_rho$$, for some homomorphism $$rho:Gamma rightarrow G$$.

This can be interpreted as a “relative” notion of a flat connection since, if $$X$$ is the universal covering space of $$Y$$ and $$Gamma=pi_1(Y)$$, then the condition for $$E$$ to be of the form $$E_rho$$ is that $$E$$ admits a flat connection.

Moreover, the same question can be extended to the holomorphic category. For example, we can take $$X$$ a Riemann surface and $$G=U(n)$$. In that situation, if $$X$$ is the hyperbolic plane and $$Y$$ is a compact Riemann surface of genus $$geq 2$$, the condition for a holomorphic Hermitian vector bundle $$E$$ to be of the form $$E_rho$$ is that it is stable of degree $$0$$ (this is the Theorem of Narasimhan-Seshadri).