  One of the key points in Galois theory via field extensions is to build up a
correspondence between subfields of a field and subgroups of its automorphism
group, so as to study fields via methods of groups. As an analogue of the
Galois theory, we want to discuss the relations between cluster subalgebras of
a cluster algebra and subgroups of its automorphism group and then set up the
Galois-like method.
  In the first part, we build up a Galois map from a skew-symmetrizable cluster
algebra $\mathcal A$ to its cluster automorphism group, and introduce notions
of Galois-like extensions and Galois extensions. A necessary condition for
Galois extensions of a cluster algebra $\mathcal A$ is given, which is also a
sufficient condition if $\mathcal A$ has a $\mathcal{D}$-stable basis or stable
monomial basis with unique expression. Some properties for Galois-like
extensions are discussed. It is shown that two subgroups $H_1$ and $H_2$ of the
automorphism group $\text{Aut}\mathcal A$ are conjugate to each other if and
only if there exists $ f \in \text{Aut}\mathcal{A} $ and two Galois-like
extension subalgebras $\mathcal A(\Sigma_1)$, $\mathcal A(\Sigma_2)$
corresponding to $H_1$ and $H_2$ such that $f$ is an isomorphism between
$\mathcal A(\Sigma_1)$ and $\mathcal A(\Sigma_2)$.
  In the second part, as the answers of two conjectures proposed in the first
part, for a cluster algebra from a feasible surface, we prove that Galois-like
extension subalgebras corresponding to a subgroup of a cluster automorphism
group have the same rank. Moreover, it is shown that there are order-preserving
reverse Galois maps for these cluster algebras. We also give examples of
$\mathcal{D}$-stable bases and some discussions on the Galois inverse problem
in this part.
