Going down for fixed-point subring under finite automorphism group

Statement
Suppose $$B$$ is a commutative unital ring and $$G$$ is a finite subgroup of the automorphism group of $$B$$. Suppose $$A = B^G$$is the fixed-point subring. Then, the extension $$B$$ of $$A$$ is a going down extension; in other words, we have going down for prime ideals. Explicitly:

If $$P_1 \supset P_2$$ are prime ideals of $$A$$ and $$Q_1$$ is a prime ideal of $$B$$ contracting to $$P_1$$ then there exists a prime ideal $$Q_2$$ of $$B$$ contracting to $$P_2$$, such that $$Q_2 \subset Q_1$$.

Facts used

 * Ring is integral extension of fixed-point subring under finite automorphism group
 * Integral extension implies surjective map on spectra
 * Automorphism group acts transitively on fibers of spectrum over fixed-point subring
 * Going up theorem

Proof outline

 * Show that $$B$$ is an integral extension of $$A$$
 * Use the fact that integral extensions give surjective maps on spectra,to find a prime ideal $$Q_2'$$ lying over $$P_2$$
 * Use going up to find a prime ideal $$Q_1'$$ lying over $$P_1$$ and containing $$Q_2'$$
 * Use the transitivity of action of $$G$$ on the fiber over $$P_1$$ to find an automorphism that sends $$Q_1'$$ to $$Q_1$$
 * The image of $$Q_2'$$ under that automorphism is the required $$Q_2$$