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On a question of Külshammer for representations of finite groups in reductive groups

Let $G$ be a simple algebraic group of type $G_2$ over an algebraically closed field of characteristic $2$. We give an example of a finite group $Γ$ with Sylow $2$-subgroup $Γ_2$ and an infinite family of pairwise non-conjugate homomorphisms $ρ\colon Γ\rightarrow G$ whose restrictions to $Γ_2$ are all conjugate. This answers a question of Burkhard Külshammer from 1995. We also give an action of $Γ$ on a connected unipotent group $V$ such that the map of 1-cohomologies ${\rm H}^1(Γ,V)\rightarrow {\rm H}^1(Γ_p,V)$ induced by restriction of 1-cocycles has an infinite fibre.