Question: Let $G=\operatorname{Aut}(\mathbb Z_4\times\mathbb Z_4)$, $c$ be an element in $G$ such that $c:(a,b)\mapsto(3a,3b)$. Prove that $\langle c\rangle\triangleleft G$, and that $$G/\langle c\rangle\cong S_4\times\mathbb Z_2.$$
My Attempt
Rephrasing the question to matrix form, $G=\rm{GL}(2,\mathbb Z_4)$, the $c$ turns into $\left(\begin{matrix}3&0\\0&3\end{matrix}\right)$. As $c$ is a diagonal matrix, $M\langle c\rangle M^{-1}=\langle c\rangle$ for all $M$. Hence $\langle c\rangle\triangleleft G$.
Note that the RHS of the desired isomorphism is a direct product, there must be an element $d$ having order $2$ in $G/\langle c\rangle$ such that it commutes with all elements in the quotient group. One problem I need to overcome is to express the elements in the quotient group. I tried to express them as cosets, but it is hard to tackle. One possible pathway is to find this $d$ and view the quotient group $G/\langle c\rangle/\langle d\rangle$ as a permutation of $4$ elements.