Automorphism group acts transitively on fibers of spectrum over fixed-point subring

Statement
Let $$G$$ be a finite group acting as automorphisms of a commutative unital ring $$B$$, and let $$A$$ be the subring of $$B$$ comprising those elements that are fixed under every element of $$G$$ (i.e. $$B^G = A$$). Consider the map at the level of spectra:

$$Spec(B) \to Spec(A)$$

There is a natural action of $$G$$ on the fibers over any point in $$A$$. The action is transitive.

Proof outline

 * Consider the $$G$$-orbit of one element in the fiber
 * Prove that any element in the fiber must, as a prime ideal, be contained in the union of all the prime ideals in that $$G$$-orbit. (This is the step where the main work is done)
 * Apply the prime avoidance lemma