Automorphism group of $C_{8}\times C_{2}$

Question

I am trying to figure how to compute ${\rm Aut}(C_2 \times C_8)$. Using GAP the answer is ${\rm Aut}(C_2 \times C_8)\simeq C_2\times D_8$.

I know that ${\rm Aut}(C_8)\simeq C_2 \times C_2$ and that ${\rm Aut}(C_2) \simeq 1 $, but that seems to have no direct relation. I would be very grateful if you could help or direct me to a relevant reference.

Thank you!