Automorphisms of a class of finite \(p\)-groups with a cyclic derived subgroup (Q2042275)

Let \(p\) be an odd prime, \(k\) a positive integer and \(G\) a finite \(p\)-group given by the central extension \(1\to G^\prime \to G \to \mathbb Z_{p^k} \times \dots \times \mathbb Z_{p^k}\) where \(G^\prime \cong \mathbb Z_{p^k}\) and \(\zeta G/G^\prime\) is a direct factor of \(G/G^\prime.\) The structure of the automorphism group of \(G\) is determined.