The fundamental groupoid functor $\Pi : \mathbf{Top} \rightarrow \mathbf{Gpd}$ is not full. However, I am interested in special cases where it is always possible to lift a morphism (functor) between fundamental groupoids to a continuous map between spaces.
In particular, is it the case that we can always lift endomorphisms $\Pi(X) \rightarrow \Pi(X)$ to continuous maps $X \rightarrow X$? If not in general, then what about for path spaces: can we lift endomorphisms $\Pi(X^I) \rightarrow \Pi(X^I)$ to continuous maps $X^I \rightarrow X^I$?
If a universal (in some sense) choice of liftings can be made, that would be cool, but I'm satisfied with simply the existence of a continuous function that maps via $\Pi$ back down to the functor in question.