The Setup:
Let $ \mathbb{F}$ be a finite field of characteristic $ p$ and $ W(\mathbb{F})$ the ring of Wittvectors with residue field $ \mathbb{F}$ . Recall that filtered Dieudonne’ $ W(\mathbb{F})$ module also known as a FontaineLaffaille module is a $ W(\mathbb{F})$ module furnished with a decreasing, exhaustive, separated filtration of submodules $ \{F^i M\}$ and for each integer $ i$ a $ \sigma$ semilinear map $ \varphi^i=\varphi_M^i:F^i M\rightarrow M$ . These maps are required to satisfy two conditions

the following compatibility relation is satisfied $ \varphi^{i+1}=p \varphi^i$ ,

$ \sum_i \varphi^i(F^i M)=M$ .
Let $ \text{MF}_{tor}^{f}$ denote the category of FontaineLaffaille modules $ M$ with morphisms satisfying the conditions alluded to above. For $ a<b$ let $ \text{MF}_{tor}^{f,[a,b]}$ let the full subcategory of $ \text{MF}_{tor}^{f}$ whose underlying modules $ M$ satisfy $ F^0 M=M$ and $ F^p M=0$ .
The FontaineLaffaille functor $ \text{U}:\text{MF}_{tor}^{f,[0,p]}\rightarrow \text{Rep}_{W(\mathbb{F})}(\text{G}_{\mathbb{Q}_p})$ where $ \text{Rep}_{W(\mathbb{F})}(\text{G}_{\mathbb{Q}_p})$ is the category of continuous $ W(\mathbb{F})[\text{G}_{\mathbb{Q}_p}]$ modules that are finitelength $ W(\mathbb{F})$ modules. It is a basic fact that if $ M$ has the structure of a free $ W(\mathbb{F})$ module of rank $ n$ (in greater detail, $ F^j M$ are all free $ W(\mathbb{F})$ modules and the maps $ \varphi^j$ and $ W(\mathbb{F})$ module maps and may be viewed as matrices with entries in $ W(\mathbb{F})$ ) then $ \rho_M:=\text{U}(M)$ is a Galois representation $ \rho_M:\text{G}_{\mathbb{Q}_p}\rightarrow \text{GL}_n(W(\mathbb{F}))$ .
Question: Let $ G\subset \text{GL}_n$ be an algebraic subgroup of $ \text{GL}_n$ defined over $ \mathbb{Q}$ (I’m mainly interested in the exceptional groups, like for instance $ G_2$ ). What condition on the matrices $ \varphi^j_M$ ensures that the image of $ \rho_M:=\text{U}(M)$ lies in $ G(W(\mathbb{F}))$ so that it is a Galois representation $ \rho_M:\text{G}_{\mathbb{Q}_p}\rightarrow G(W(\mathbb{F}))$ .
Comment: For the classical groups like $ \text{GSp}_{2n}$ I’m aware how to do this. It involves making use of the alternating form and the functoriality of $ U$ .