  We show for all local fields $K/\mathbb{Q}_p$, with $p >3$, all
representations $\bar\rho:G_K \to G_2(\bar{\mathbb{F}}_p)$ admit a crystalline
lift $\rho: G_K\to G_2(\bar{\mathbb{Z}}_p)$, where $G_2$ is the exceptional
Chevalley group of type $G_2$.
  The main ingredient is a new technique for analyzing the obstruction of
non-abelian extensions of Galois modules, which has roots in combinatorial
group theory. We also rely on the Emerton-Gee stack of $(\phi, \Gamma)$-modules
to construct abelian extensions.
