Let $X,Y$ be compact Hausdorff spaces, fix a continuous surjection $f\colon X\to Y$.
Suppose $B\subseteq Y$ has the property that $f^{-1}[B]$ is Borel. Is $B$ necessarily Borel?
Note that if $X$ is metrisable, then this is true by Suslin's theorem, as then $f^{-1}[B]$ is analytic, and its complement is analytic as well, which easily implies that the same is true about $B$.
Since continuous maps between compact Hausdorff spaces are closed, it is true if the preimage is $F_\sigma$, and this implies the same for $G_\delta$ by complementation. So in a counterexample, we would need $f^{-1}[B]$ to be at best $F_{\sigma\delta}$ (or $G_{\delta\sigma}$).
I think it should be false in general, but I see no obvious counterexample.