Summary: Let \(C_{p^m}\) be a cyclic group of order \(p^m\), where \(p\) is a prime and \(m\) is an arbitrary positive integer. It is proved that the wreath product \((\ldots((C_{p^m}\mathrm{Wr}C_{p^m})\mathrm{Wr}C_{p^m})\mathrm{Wr}\ldots)\mathrm{Wr}C_{p^m}\) (\(n\) factors) is determined for each natural \(n\) by its endomorphism semigroup in the class of all groups. It follows that (a) every Sylow subgroup of a finite symmetric group is determined by its endomorphism semigroup in the class of all groups, (b) each finite \(p\)-group \(G\) is embeddable into a finite \(p\)-group \(G^\ast\) such that \(G^\ast\) is determined by its endomorphism semigroup in the class of all groups.