The functor from schemes to sets, given by taking the set of closed points, is neither full nor faithful. For example, the spectrum of a field with nontrivial automorphisms has a single closed point, but its automorphism group is nontrivial. On the other hand, the complex projective line has automorphism group $PGL_2(\mathbb{C})$, but the set of closed points is an abstract set of cardinality $2^{\aleph_0}$, and has an automorphism group of strictly larger cardinality. What you can say is that the functor induces a homomorphism (that may be neither injective nor surjective).
If you want to consider automorphism groups of base extensions, you might as well assemble them into the automorphism group sheaf $\underline{\operatorname{Aut}} (S)$, which eats a scheme $T$, and returns the automorphism group of $S \times T$ . (as a $T$-scheme). When $S$ is projective, this sheaf is represented by a scheme, whose $k$-points are precisely the automorphisms of $S_k$. For example, $\mathbb{P}^1$ has automorphism group scheme $PGL_2$.
The functor from schemes to sets, given by taking the set of closed points, is neither full nor faithful. For example, the spectrum of a field with nontrivial automorphisms has a single closed point, but its automorphism group is nontrivial. On the other hand, the complex projective line has automorphism group $PGL_2(\mathbb{C})$, but the set of closed points is an abstract set of cardinality $2^{\aleph_0}$, and has an automorphism group of strictly larger cardinality. What you can say is that the functor induces a homomorphism (that may be neither injective nor surjective).
If you want to consider automorphism groups of base extensions, you might as well assemble them into the automorphism group sheaf $\underline{\operatorname{Aut}} (S)$, which eats a scheme $T$, and returns the automorphism group of $S \times T$. When $S$ is projective, this sheaf is represented by a scheme, whose $k$-points are precisely the automorphisms of $S_k$. For example, $\mathbb{P}^1$ has automorphism group scheme $PGL_2$.