Let $f:X\to Y$ by a continuous function, where $X,Y$ are metric spaces. Let $\mathcal B(X)$ be the Borel sigma-field on $X$. Consider the sigma-field $\{B\in \mathcal B(X):B=f^{-1}(f(B))\}$ on $X$ that consists of the invariant Borel sets. Is this sigma field equal to $\{f^{-1}(A): A\in \mathcal B(Y)\}$?
It is easy to show that the latter is included in the former.
Update: Assume that $X$ and $Y$ are complete and separable.
Let $\mathbb R$ be the underlying set of $X$ and $Y$, let $X$ be equipped with discrete topology and $Y$ with usual topology. Both are metrizable.
Let $f$ be the identity function and observe that the function is continuous.
Then the first collection is $\wp (\mathbb R) $ and the latter is $\mathcal B (\mathbb R) $ hence is a proper subcollection of the first.