Quotients of \(G\)-stable closed subschemes, Cartesian diagrams, and closed immersions (Q1915728)

Let \(T\) be a separated scheme of finite type over an algebraically closed field \(k\) with a finite group \(G\) acting on \(T\), and let \(S\) be a \(G\)-stable closed subscheme of \(T\). Suppose the (categorical) quotients \(T/G\) and \(S/G\) exist. Is then the morphism \(i/G: S/G \to T/G\) induced by the closed immersion \(i:S \to T\) a closed immersion? Furthermore, is \(S\) isomorphic to a cartesian product \((S/G,i/G) \times_{T/G} (T,\pi_T)\), where \(\pi_T:T\to T/G\) is the quotient morphism? In the present article, these questions are answered affirmatively if the \(G\)-action on \(T\) is free. It is also shown that the answer to the second question is negative if the \(G\)-action is not free.