ag.algebraic geometry - algebraic group G vs. algebraic stack BG

If G is a group scheme over k (algebraic closed), then let me talk through how to get G back by looking at the stack BG. The k points of BG (which is a groupoid) consist of one point whose automorphisms are the k points of G. The pullback of this point to Spec A for any k-algebra A has automorphisms given by the A points of G. If you think of BG points as principal bundles, I'm saying the automorphsims of the trivial bundle on Spec A are the A points of the group.

So what happens if you deform BG? You still have this one point, you can't deform that to anything, so you can only change its morphisms. That's your G' (you get an algebraic group since you can pullback to all the Spec A's). How you see it's BG' is a little trickier, so maybe I should leave it to a real algebraic geometer, but I think that the idea is that BG is distinguished by being the sheafication of the trivial bundles in the smooth/fppf topology, and this won't change when you deform.