Consider smooth complex varieties $X$ and $Y$ such that there is a finite etale morphism $p:X\rightarrow Y$. Let $Z$ be a smooth subvariety of $Y$. We have the morphism of fundamental groups :
$$p_*:\pi_1(X)\rightarrow \pi_1(Y)\ \text{and}$$ $$i_*:\pi_1(Z)\rightarrow\pi_1(Y)\,.$$
Suppose there is a morphism $g:\pi_1(Z)\rightarrow \pi_1(X)$ such that $p_*\circ g=i_*$, then what does this say?
Does it mean that $p^{-1}(Z)$ is a reducible subvariety of $X$?
Edit - I want to assume that the degree of the finite map is $\geq 2$.