The second cohomology group for the extensions of the group \(\text{Aut }C_{12}\) by the cyclic group \(C_{12}\) (Q1333425)

Let \(C_{12}\) be a cyclic group of order 12, identified with \(\mathbb{Z}/12\mathbb{Z}\). Then \(\text{Aut}(C_{12}) = \{\tau_ 1, \tau_ 5, \tau_ 7, \tau_{11}\}\), where \(\tau_ r (j) = rj \pmod{12}\) for \(r \in \{1, 5, 7, 11\}\) and all \(j \in C_{12}\), and the mapping \(\Delta\) of \(\Aut (C_{12})\) into itself defined by \(\Delta (\tau_ 1) = \tau_ 1\), \(\Delta(\tau_ 5) = \tau_{11}\), \(\Delta (\tau_ 7) = \tau_{11}\), \(\Delta (\tau_{11}) = \tau_ 1\) is an operator of \(\Aut (C_{12})\) on \(C_{12}\). The author determines the cohomology group \(H^ 2_ \Delta (\Aut (C_{12}), C_{12})\) (which is a group of order 8) using MacLane's method based on free groups [\textit{S. MacLane}: Ann. Math., II. Ser. 50, 736-761 (1949; Zbl 0039.257)].