Hochster-Eagon type theorem for Serre's $(S_n)$ condition

Let $(A,\mathfrak{m})\to (B,\mathfrak{n})$ be a pure homomorphism between Noetherian commutative rings. If $B\mathrm{/}\mathfrak{m}B$ is an Artinian ring, then we have $\dim A=\dim B$ and $A\ge B$. Using this version of Hochster-Eagon theorem, we prove the following: Let $A\to B$ be a pure homomorphism between Noetherian commutative rings. Assume that the fiber ring $\kappa (\mathfrak{p}){\otimes }_{A}B$ is Artinian for each $\mathfrak{p}\in A$, and $B$ satisfies Serre’s $(S_n)$ condition. Then $A$ also satisfies Serre’s $(S_n)$ condition. In particular, if a finite group $G$ acts on $B$ and the order $\mathrm{\mid }G\mathrm{\mid }$ of $G$ is invertible in $B$, and if $B$ is Noetherian with the $(S_n)$ condition, then the ring of invariants $A=B^G$ also satisfies the $(S_n)$ condition.