Suppose we are given an open cover $\mathcal{U}=(U\_{i})\_{i \in I}$ of a smooth manifold $M$, a cocycle of smooth transition functions $g\_{ij}: U\_{ij} \to G$ where $G$ is a Lie group, and a (not necessarily faithful) action $\lambda: G \times S \to S$ on a smooth manifold $S$. Then this is the data of a fibre bundle $E \to M$ with typical fibre $S$ and with structure group $G$ over $M$. From this cocycle point of view, how can we describe the jet bundle $J^r E \to M$? 



Constructing jet bundles from a cocycle of smooth transition functions
You start with the trivializations $S \times U\_i \to U\_i$ over your cover. Let me presume that you can take for granted the construction of the jet bundles $J^r(S \times U\_i) \to U\_i$, whose typical fiber I'll take to be $S^r$. The group action $(\lambda,\mathrm{id}) \colon G \times (S\times U\_i) \to S \times U\_i$ has the jet extension
$$
 (\lambda^r,\mathrm{id}) = j^r(\lambda,\mathrm{id}) \colon
 (G \times S^r) \times U\_i = G \times J^r(S\times U\_i) \to S^r \times U\_i = J^r(S \times U\_i) .
$$
You already have your $G$-cocycle on $M$, so you need only use the typical fiber $S^r$ and the action $\lambda^r \colon G \times S^r \to S^r$, instead of $S$ and $\lambda$, to get $J^rE \to M$.


