Let ppTop denote the category of pointed and path connected topological spaces with morphisms base-preserve continuous maps. The fundamental group gives a functor $FG: \text{ppTop}\to \text{Gp}$ where GP is the category of groups.

Now we consider the category pTop consisting of path-connected topological spaces and we can naturally define the fundamental groupoids instead of fundamental groups on pTop. If we want to define the fundamental group then we need to choose a base point. Notice that there is a forgetful functor $\text{For}:\text{ppTop}\to \text{pTop}$.

>>My question is: could we lift the functor $FG: \text{ppTop}\to \text{Gp}$ to a functor $\widetilde{FG}: \text{ppTop}\to \text{Gp}$ such that $\widetilde{FG}\circ \text{For}=FG$. If not, how to construct a contradiction?