If $f,g:X\to Y$ are homotopic maps, are the images of $f_*$ and $g_*$ isomorphic? Let $f,g:X\to Y$ be homotopic maps, and let $f_*, g_*$ be the maps induced on the fundamental groups. 
Are the images of $f_*$ and $g_*$ isomorphic? In other words, is it true that $f_*(\pi_1(X,x_0))\cong g_*(\pi_1(X,x_0))$?
If not, can someone give a counter-example? This point is not explicitly addressed in the book that I'm reading 
 A: If $X$ and $Y$ are pointed spaces and $f$ and $g$ are pointed maps, homotopic as pointed maps. They're not just isomorphic, they're equal. If $f\simeq g$, then $f_* = g_*$. 
Otherwise, let $f\simeq g$. Let $F(x,t):X\times [0,1]\to Y$ be such a homotopy. Let $p:[0,1]\to X$ be a path in $X$ that is a loop at $x_0$ (i.e. $p(0)=p(1)=x_0$). Then by definition, $f_* [p] :=[f\circ p]$ is a loop based at $f(x_0)$ and $g_*[p]=[g\circ p]$ is a loop based at $g(x_0)$. Then $F(x_0,t)$ defines a path from $f(x_0)$ to $g(x_0)$, and conjugating $F(p(t),1)=g_*[p]$ by this path gives a path homotopic to $F(p(t),0)=f_* [p]$. Thus the groups are not just isomorphic, they are conjugate.
If I'm a little more careful with my variables, I can give the explicit homotopy. Let $p'_{t_1}(t): [0,1]\times [0,1]\to Y$ be the family of paths
$p'_{t_1}(t)=F(x_0,t_1t)$ from $f(x_0)$ to $F(x_0,t_1)$. Then consider the family of loops $\ell_{t_1}(t)=F(p(t),t_1)$ which are based at $F(x_0,t_1)$. Thus we can compose the loops as follows $p_{t_1}^{\prime-1} \ell_{t_1}p'_{t_1}$ to get a loop $\gamma_{t_1}$ from $f(x_0)$ to $f(x_0)$. Then $\gamma_0 = f_*[p]$ and $\gamma_1$ is precisely the conjugate of the loop $g_*[p]$.
