This is impossible if $f$ is invertible, without further assumptions such as bijective, differentiable, etc. Let $Q_1,Q_2$ be probability measures on a measurable space $(\Omega, \mathcal{F})$, and assume $f_* Q_1 = f_* Q_2$ for some invertible (bimeasurable) $f : (\Omega,\mathcal{F}) \to (\Xi,\mathcal{G})$. For any $A\in \mathcal{F}$, definitions give
$$ Q_1(A) = f_* Q_1 (f(A)) = f_* Q_2 (f(A)) = Q_2(A). $$
Thus, $Q_1 = Q_2$.