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 $\mathrm{\Gamma }$ with Sylow $2$-subgroup ${\mathrm{\Gamma }}_2$ and an infinite family of pairwise non-conjugate homomorphisms $\rho \text{\hspace{0.17em}⁣}:\mathrm{\Gamma }\to G$ whose restrictions to ${\mathrm{\Gamma }}_2$ are all conjugate. This answers a question of Burkhard K"ulshammer from 1995. We also give an action of $\mathrm{\Gamma }$ 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.