Let $X$ and $Y$ be non compact complex manifolds and $f:X\to Y$ be a holomorphic fiber bundle with fibers $F$ such that $f^*:\pi_1(X)\to\pi_1(Y)$ is injective and let for any $f_1,f_2\in F$ there exists a biholomorphic map $\psi:F\to F$ such that $\psi(f_1)=f_2$. My question is $\pi_1(X)$ a normal subgroup of $\pi_1(Y)$? If not can we put some extra conditions(apart from finite fibers) such that $\pi_1(X)$ is a normal subgroup of $\pi_1(Y)$?

Any comments or suggestions on how to think about this question is highly appreciated. Thanks in Advance!!

Let $X$ and $Y$ be non compact complex manifolds and $f:X\to Y$ be a holomorphic fiber bundle with fibers $F$ such that $f^*:\pi_1(X)\to\pi_1(Y)$ is injective and let for any $f_1,f_2\in F$ there exists a biholomorphic map $\psi:F\to F$ such that $\psi(f_1)=f_2$ and $\psi\in Deck(X/Y)$. My question is $\pi_1(X)$ a normal subgroup of $\pi_1(Y)$? If not can we put some extra conditions(apart from finite fibers) such that $\pi_1(X)$ is a normal subgroup of $\pi_1(Y)$?

Any comments or suggestions on how to think about this question is highly appreciated. Thanks in Advance!!