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 (measurable) $f : \Omega \to \Xi$.  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, $Q_1 = Q_2$.