Thank you so much for your answer. Your explanation of the Thomason-Trobaugh theorem and its extension to the regular case has been beneficial for understanding the extension problem. I have a more specific question, Consider given a morphism $E\to \mathcal F$ on $X_K$ and an extension of E to E' on X, what is the possible obstruction to extend $E\to F$ to some $E'\to F'$ (finding F' together with a morphism)?

@DEBAJYOTI DE , thanks for pointing out, by B_kG I mean the classifying stack BG over Spec k, parametrizing G-torsors over k in fppf topology. (I think here étale topology is enough)