Wreath products of cyclic \(p\)-groups as automorphism groups. (Q1770517)

The authors prove the following statement: The wreath product \(A\text{\,wr\,}B\) with cyclic \(p\)-groups \(A\) and \(B\) is the automorphism group of a group \(G\) iff \(A\) is of order 2 and \(B\) is of order 2 or 4. The group \(G\) is in the first case \(D_8\) or \(C_4\times C_2\) and in the second case some infinite nilpotent group of class 3, the possibilities are described completely.