On perfectly meager sets in the transitive sense (Q2758999)

A~set \(X\subseteq 2^\omega\) is said to be perfectly meager in the transitive sense if for any perfect set \(P\subseteq 2^\omega\) there is an \(F_\sigma\)~set~\(F\) containing~\(X\) such that for every \(t\in 2^\omega\), \((F+t)\cap P\) is meager in the relative topology of~\(P\). The author proves that assuming \(\mathfrak c\leq\aleph_2\) there is a~perfectly meager set (even universally meager) which is not perfectly meager in the transitive sense.