I am working on a problem and at some point i wanted to check if there are any homomorphisms (non-trivial) from $Z_{7}$ to $\operatorname{Aut}( Z_{4} \times Z_{2})$ I was hoping there won't be any but i got the automorphisms explicitly ( 8 of them) and I found 5 such homomorphisms. Is that right? Or do i have a mistake somewhere?
Thanks