On a question of K\"ulshammer 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 $\Gamma$ with Sylow $2$-subgroup $\Gamma_2$ and an infinite family of pairwise non-conjugate homomorphisms $\rho\colon \Gamma\rightarrow G$ whose restrictions to $\Gamma_2$ are all conjugate. This answers a question of Burkhard K\"ulshammer from 1995. We also give an action of $\Gamma$ on a connected unipotent group $V$ such that the map of 1-cohomologies ${\rm H}^1(\Gamma,V)\rightarrow {\rm H}^1(\Gamma_p,V)$ induced by restriction of 1-cocycles has an infinite fibre.