Let $(\mathbb{E}, \pi, \mathbb{B}, \mathbb{F})$ and $(\mathbb{E}', \pi', \mathbb{B}, \mathbb{F})$ be two $C^r$ (where $r \ge 1$) equivalent Fibre Bundles, i.e., there is a $C^r$ diffeomorphism $H: \mathbb{E} \longrightarrow \mathbb{E}'$ such that $\pi = \pi' \circ H$.
Suppose that $(\mathbb{E}, \pi, \mathbb{B}, \mathbb{F})$ has discrete structure group.
Does $(\mathbb{E}', \pi', \mathbb{B}, \mathbb{F})$ also have discrete structure group?