On a characterisation of a class of probability distributions by those of some statistics

Let $\mathscr P$ be a class of probability distributions of random element $X$. The question is whether there exist a statistic $Y=f(X)$ possesing two following properties: 1$^\circ$. Its distribution $Q_P^Y$ under $P\in\mathscr P$ does not depend on $P$, $Q_P^Y=Q_\mathscr P^Y$. 2$^\circ$. If for some $P'$ $Q_{P'}^Y=Q_\mathscr P^Y$ then $P'\in\mathscr P$. The question is solved positively for some special families $\mathscr P$.