Let $X$ be a topological space, and $x_0 \in X$. There is a natural group homomorphism $\pi_1(X, x_0) \to H_1(X)$ from the fundamental group of $X$ with basepoint $x_0$ to the homology group $H_1(X)$, given by mapping a loop based at $x_0$ to its homology class.
I would like to know if it is possible to extend this to a functor $\Pi_1(X) \to H_1(X)$ from the fundamental groupoid to $H_1(X)$, viewed as a category on one object.
Is there maybe a nice reference where I could read about this?