Exercise 22, § 5

Let $A = \mathbf{Z}/2\mathbf{Z}$, $B = \mathfrak{S}_3$ and $G = A \times B$. If $s$ is an element of $B$ of order 2, let $\phi$ be the endomorphism of $G$ whose kernel is equal to $B$ and whose image is $\{1, s\}$. Show that the centre of $G$ is not stable under $\phi$.