On a problem of Zambakhidze–Smirnov

We say that the action extension problem is solvable for a bicompact group $G$ if for any metric $G$-space $\mathbb X$ and for any topological embedding $c$ of the orbit space $X$ into a metric space $Y$ there exist a $G$-space $\mathbb Z$, an invariant topological embedding $b\colon X\to\mathbb Z$, and a homeomorphism $h\colon Y\to Z$ such that the diagram
$$ </nomathmode><mathmode>
\begin{alignedat}{2} &\mathbb X\ \xrightarrow{\hskip13mm b\hskip13mm}&&\ \mathbb Z \\ {\scriptstyle p}&\downarrow\hskip30pt&&\downarrow{\scriptstyle p} \\ &X \xrightarrow{\quad c\quad} \ Y\ \xrightarrow{\quad h\quad} &&\ Z. \end{alignedat}
$$
</mathmode><nomathmode> is commutative. We prove the following theorem: for a bicompact zero-dimensional group $G$, the action extension problem is solvable for the class of dense topological embeddings.