Lavrentiev's Theorem. Suppose $X$ and $Y$ are complete metric spaces, $A\subseteq X$, $B\subseteq Y$, and $f:A\to B$ a homeomorphism. Then $f$ can be extended to a homeomorphism $\overline f :G\to H$, where $G\supseteq A$, $H\supseteq B$ are $G_\delta$ sets.
Are there any similar results for compact Hausdorff spaces, maybe replacing $G_\delta$ with $G_\kappa$ for some other infinite cardinal $\kappa$ related to the space?
I am just looking for any results on extending homeomorphisms between subspaces of a given space.