Let $f:X\rightarrow Y$ be a locally trivial fibration with a variety $F$ as the fiber. Here $X, Y, F$ are smooth, projective varieties.

Does any automorphism of $F$ induce an automorphism of $X$?

In other words, does there exist an injective group homomorphism

$$Aut(F)\rightarrow Aut(X)$$

centerof $Aut(F)$ to $Aut(X)$. – John Pardon Mar 4 at 0:42