Suppose that for any two points $x,y \in X$, all path classes from $x$ to $y$ induce the same isomorphism $\pi(X,x) \rightarrow \pi(X,y)$. From the last exercise, we see that any two paths in the space $\gamma_1, \gamma_2$ (between any points because $x$ and $y$ are arbitrary), must have the property that $\gamma_2 \gamma_1^{-1}$ is in the center of $\pi(X,x)$. But, any loop in $\pi(X,x)$ can be considered as the product of a path to some other point $y$ and its inverse. So, every member of $\pi(X,x)$ is in the center. That is, the fundamental group of $X$ must be abelian.