 \rightarrow  H^0(E, T_E(E)) \rightarrow \mathbb{H}^0(B',L_{(f',i')}^\vee) \rightarrow  \mathbb{H}^0(B, L_{(f,i)}^\vee ) \rightarrow 0 \\   0 \rightarrow \mathbb{H}^0(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^0(B,  L_{(f,i)}^\vee ) \rightarrow 0\end{aligned}$$ So the canonical map from the Lie algebra of infinitesimal automorphisms of $\zeta'$ to the Lie algebra of infinitesimal automorphisms of $\zeta$ is surjective with $0$-dimensional kernel, the canonical map from the space of first order deformations of $\zeta'$ to the space of first order deformations of $\zeta$ is surjective with $0$-dimensional kernel, and the obstruction space of $\zeta'$ equals the obstruction space of $\zeta$.  