TITLE: In a path connected space, when does every path between $p$ and $q$ induce the same isomorphism between Fundamental Groups based at $p$ and $q$?
QUESTION [5 upvotes]: Note: I will concatenate paths from left to right.
If $X$ is a path-connected topological space, and $p, q\in X$ are distinct, then any path $\gamma$ between $p$ and $q$ induces an isomorphism between $\pi_1(X, p)$ and $\pi_1(X, q)$, where a loop $\alpha$ based at $p$ is send to the loop $\gamma^{-1}\alpha\gamma$, which is based at $q$. If $\delta$ is another path between $p$ and $q$, then $\gamma$ and $\delta$ would induce the same isomorphism iff $\gamma^{-1}\alpha\gamma$ is homotopic (as a loops with fixed base point $q$) to $\delta^{-1}\alpha\delta$ for all $\alpha$, since these would then be in the same equivalence class and thus equal in $\pi_1(X, q)$. This would only seem to hold if every loop from $p$ to $q$ and then back to $p$ in the reverse direction (i.e., all paths of the form $\gamma^{-1}\gamma$ are homotopic. 
Now, this condition is enough to answer the question, but I was wondering if it was equivalent to some simpler condition. I guess it's equivalent to all paths from $p$ to $q$ being homotopic to each other (while fixing $p$ and $q$). Is there a name for this condition?

REPLY [2 votes]: One might as well state the algebraic result (as in Topology and Groupoids):
6.3.2  Let $x, x ′$ belong to the same component of the groupoid $G$. Then $a_*=b_*:  G(x) \to G(x ′ )$  for all $a, b : x → x ′$  if and only if $G(x)$ is abelian.