Linear extensions of Baire-one and Borel functions

Let $X$ and $Y$ be the Hausdorff topological spaces and let $A$ be both an $\text{fs}$- and $\text{gd}$- subset of $X$. Let also $f\text{cn}A\to Y$ be a function for which the inverse image of every open subset $U\subset Y$ is $\text{fs}$ in $X$. We show that $f$ can be linearly extended to a function with the same property defined on $X$. A similar result is proved for Baire-one function defined on an analogous subset of $\text{mR}$. We give also an answer when the extension map is (with a supremum norm) an isometry.