This is Problem 7-9 in Lee's Introduction to Topological Manifolds:
Suppose $X$ and $Y$ are connected topological spaces, and the fundamental group of $Y$ is abelian. Show that if $F,G: X \rightarrow Y$ are homotopic maps such that $F(x) = G(x)$ for some $x\in X$, then $F_*=G_* : \pi_1(X,x) \rightarrow \pi_1(Y,F(x))$. Give a counterexample to show that this might not be true if $\pi_1(Y)$ is not abelian.
I can't solve this problem.
First of all, I wonder why connectedness condition is given not path connecteness condition. If $Y$ is not path connected, we can not say, I think, THE fundamental group of $Y$.
Let $H:X \times I \rightarrow Y$ be a homotopy of $F$ and $G$. To prove $F_*=G_*$, put $[f]\in \pi_1(X,x)$ and we must show that $F \circ f \sim G \circ f$. $H(f(s),t)$ is not the desired one since the end points is not fixed. Putting $h(t) = H(f(0),t)$ and $k(t) = H(f(1),t)$, we can say $(F \circ f) \cdot k \sim h \cdot (G \circ f)$ by the square lemma. I can not progress at this point. Could I have any hint?
If possible, I would like to have the counter example.