Let $g: A \to B$ be a local ring morphism between local Noetherian (commutative) rings $A,B$ (so $g(m_A) \subset m_B$ for the unique maximal ideals of the corresponding rings). Assume that the induced map of completions of the two local rings $\widehat{g}: \widehat{A} \to \widehat{B} $ is an isomorphism.

What do I then know about the previous morphism $g: A \to B$ and the correspondence of the maximal ideals $m_A$ and $m_B$? So essentially which information can we derive/ "pull back" about $g,A$ and $B$ from knowing that $\widehat{g}$ is an isomorphism?

I know that we can deduce following informations instantly:

the canonical maps $A/m_A^n \cong \widehat{A}/\widehat{m_A}^n \to \widehat{B}/\widehat{m_B}^n \cong B/m_B^n $ are surjective for all $n \in \mathbb{N}_{\ge 1}$

$\widehat{m_B}=\widehat{m_A}\widehat{B}$

for every $k$ there exist a $d_k$ with $ g(m_A)^{d_k} \subset m_B ^k$ and vice versa (so same topology)

The question is what do we know about $g:A \to B$ (injective, surjective? ... by a bunch of counterexamples we know that $g$ is almost never an isomorphism, but what "can nevertheless be saved"?) and how are related the maximal ideals $m_A$ and $m_B$?

When we can expect $g(m_A)B=m_B$?

What do we still know about $g$ if we neglect the Noetherian condition?

Remark: This question arises from following former MathSE question of mine and intends to generalize how the "tool box" of completions can be progressively applied to deduce some useful informations/relations about the initial morphism of local rings.