Does the classifying map of a fibre bundle only depend on the transition functions?
Precisely,
Let $\xi$ and $\eta$ be two fibre bundles over $B$, whose transition functions are same, both of their structure groups are $GL(\mathbb{F}^n)$, for $\mathbb{F}=\mathbb{R},\mathbb{C}$ or $\mathbb{H}$. The fibres of $\xi$ and $\eta$ may be different.
Let $f:B\to G_n(\mathbb{F}^\infty)$ be the classifying map of $\xi$. Let $g:B\to G_n(\mathbb{F}^\infty)$ be the classifying map of $\eta$.
Does $f=g$?