$\require{AMScd}$I am currently thinking about [(strict) henselisations](https://stacks.math.columbia.edu/tag/0BSK) but I don't know too much literature about the topic. So I am wondering if there is a natural way to restrict maps between strict henselisations to henselisations:

Let $A$, $B$ be local rings with an injective homomorphism $h:A\to B$. If I have a homomorphism $f:A^\text{sh} \to B^\text{sh}$, is there always a homomorphism $g:A^\text h \to B^\text h$ such that the following diagram commutes?

\begin{CD}
A^\text{sh} @>f>> B^\text{sh}\\
@AAA @AAA\\
A^\text h @>>g> B^\text h
\end{CD}

Equivalently, I am asking if you start with $x \in A^\text h \subset A^\text{sh}$, is $f(x) \in B^\text h$? It feels like this should be related to Galois theory, where Galois extensions fix the base field.

(Note: There might be requirements on $A$ and $B$ like being normal, or $B$ being a field. I'm also interested in answers with more assumptions than stated above.)

**Clarifications:**
The underlying injective homomorphism $h: A \to B$ is *not necessarily a local map* but $f$ commutes with $h$.

\begin{CD}
A^\text{sh} @>f>> B^\text{sh}\\
@AAA @AAA\\
A @>>h> B
\end{CD}

The example I have in mind is the following: Pick a ring $R$ and a maximal ideal $\mathfrak{m} \in R$ and a map $\operatorname{Frac}(R) \to \operatorname{Frac}(R)^\text{sep}$ (i.e. a geometric point over the generic point of my curve $\operatorname{Spec}(R)$). Then $A := R\_{\mathfrak{m}}$, $B:=\operatorname{Frac}(R)$, the map $h: A \to B$ is not local and $B \to B^\text{sh} = \operatorname{Frac}(R) ^\text{sep}$ is given by above chosen geometric point.