For any such lift $\widetilde{FG}:\mathrm{pTop}\to \mathrm{Gp}$ the induced map $pTop(X,Y)\to Gp(\widetilde{FG}(X),\widetilde{FG}(Y))$ must factor through the set of homotopy classes of the maps between $X$ and $Y$, for any spaces $X,Y$.
Indeed, consider the maps $\iota_0,\iota_1:X\rightrightarrows X\times[0,1]$ given by $\iota_a(x)=(x,a)$ for $a=0,1$. The morphisms $\iota_0,\iota_1$ are images of the morphism $\tilde\iota_0:(X,x)\to (X\times[0,1],(x,0))$ and $\tilde\iota_1:(X,x)\to (X\times[0,1],(x,1))$ in the pointed category. Since the functor $FG$ carries $\tilde\iota_0$ and $\tilde\iota_1$ to isomorphisms, so does the lift $\widetilde{FG}$ to the morphisms $\iota_0,\iota_1$. Analogously, the projection $p:X\times[0,1]\to X$ gets sent to an isomorphism. Since $p\circ\iota_0=p\circ\iota_1=id_X$, the induced morphisms $\widetilde{FG}(\iota_0),\widetilde{FG}(\iota_1):\widetilde{FG}(X)\to \widetilde{FG}(X\times[0,1])$ must be equal. In particular, the two compositions $$pTop(X\times[0,1],Y)\rightrightarrows pTop(X,Y)\to Gr(\widetilde{FG}(X),\widetilde{FG}(Y))$$ are equal which implies that any two homotopic maps $f_0,f_1:X\to Y$ induce the same maps between $\widetilde{FG}(X)$ and $\widetilde{FG}(Y)$.
Take now $X=S^1$ equipped with a base point $p\in S^1$. For any $(Y,y)\in ppTop$ the map induced by $FG$ sends a pointed morphism to its homotopy class$$ppTop((S^1,p),(Y,y))\twoheadrightarrow Gp(\mathbb{Z},\pi_1(Y,y))=\pi_1(Y,y)$$
By the above observation, a lifting $\widetilde{FG}$ would yield a factorization of this map through the set of homotopy classes of unpointed maps $S^1\to Y$. However, the latter set is identified with the singular homology group $H_1(Y,\mathbb{Z})$, so taking $Y$ to be any space with a non-abelian fundamental group brings us to a contradiction.