Change of rings of scalars and projecivity?

Question

Let $f:R\to S$ be a ring homomorphism and M be a left S-module. We can consider M as an R-module via $ r.m := f(r)m $. I know that if M is a flat S-module and S is flat as R-module then M is a flat R-module. My question is about projectivity and finite presentation, what we can say?

Answer 1

Consider $f\colon\mathbb{Z}\to\mathbb{Q}$. In this case we're just considering $\mathbb{Q}$-vector spaces as $\mathbb{Z}$-modules and clearly projectivity is not preserved, because no nontrivial $\mathbb{Q}$-vector space is a projective $\mathbb{Z}$-module.

Even being finitely generated is not preserved in the same situation.