I'm working on the classification on the group of order 75. Part of its question is to decide the homomorphism between $\Bbb Z_3$ and $\operatorname{Aut}(\Bbb Z_5\times \Bbb Z_5)$, which is to find the elements in $\operatorname{Aut}(\Bbb Z_5\times \Bbb Z_5)$ with order 3.
Since $\lvert\operatorname{Aut}(\Bbb Z_5\times \Bbb Z_5)\rvert=24\times 20$, it is messy to discuss the Sylow $3$-subgroups based on the order, also, direct computation could do, but it does not seem a good way. I wonder if there's any easier way to determine it.