Let $X = \mathbb{C} \setminus \{ \pm 2 \}$ and $Y = \mathbb{C} \setminus \{ \pm 1, \pm 2 \}$. The map $$ p : Y \to X : z \mapsto z^3 - 3z $$ is a 3-branched covering as given in this question of Math Student 020.
Question: What is the induced homomorphism $p_*:\pi_1(Y)\rightarrow\pi_1(X)$?
My try: Let $a$, $b$, $c$, $d$ be loops around $-2,-1,1,2$ which generate $\pi_1(Y)=F(a,b,c,d)$ and $u,v$ be loops around $-2,2$ which generate $\pi_1(X)=F(u,v)$ then $p_{*}(a)=u^3$, $p_*(b)=uv^2$, $p_{*}(c)=u^2v$, $p_{*}(d)=v^3$. Is it wrong?