I don't think that the following is known, but before going to other things, I would like to know what can be said about it. Thanks in advance for any relevant comment !

So here is the situation. Let $k'\subset k$ be a finite field extension. Take affine algebraic group schemes defined over $k'$ together with a central isogeny $\pi : \tilde{G}\rightarrow G$ (i.e. $\pi $ is surjective on underlying topological space, and is a finite flat morphism such that $ker$ $\pi\subset Z(G)$). 

Given a subgroup $\Gamma \leq \tilde{G}(k)$, assume that $\Gamma _{down}=\pi_{k}(\Gamma )$ lies in fact already in $G(k')$. Can we conclude that $\pi_{k'}^{-1}(\Gamma_{down})$ is big enough compared to $\Gamma $ (more precisely, that its intersection with $\Gamma $ is of finite index in $\Gamma $ ?)

Ok, now more precision about assumption (but I would be interested to have counter-examples if not in that situation, especially for the first assumption) :  
1) $k$ is a local field, and $k'$ a closed subfield  
2) $\tilde{G}$ (resp. $G$) is absolutey simple simply connected (resp. adjoint)  

Also, let me stress that I am mainly interetsed in the positive characteristic case, i.e. $k$ a finite extension of $\mathbb{F}_{p}$((T)), and that I do not assume that the extension is Galois (but I would be interested to know what Galois cohomology can bring to the matter, I'm not at all familiar with that theory).