Suppose that $K/\mathbb{Q}_l$ is a finite extension, with ring of integers $\mathcal{O}_K$. Suppose $\mathcal{G}/K$ is a (linear) algebraic group (connected+reductive), and $\Gamma\subset \mathcal{G}(K)$ is a hyperspecial maximal compact subgroup. This just means (I believe) that we can find $\tilde{\mathcal{G}}/\mathcal{O}_K$ with $\tilde{\mathcal{G}}_K=\mathcal{G}$ and $\tilde{\mathcal{G}}(\mathcal{O}_K)=\Gamma\subset\tilde{\mathcal{G}}(K)$.

Suppose we have an automorphism $\alpha$ of $\mathcal{G}/K$ preserving $\Gamma=\tilde{\mathcal{G}}(\mathcal{O}_K)$. Can we necessarily extend $\alpha$ to an automorphism of $\tilde{\mathcal{G}}$? If not, are there any nice conditions under which we can? For instance, what if $\mathcal{G}$ is simple or even simple and simply connected?

Edit: added connected, reductive hypotheses on $\mathcal{G}$, which were there in my head, but I forgot to write them...