This is a part in Dummit & Foote's abstract algebra on page 184.
In the middle of the example of groups of order $30$, the book lets $H=\langle a\rangle \times\langle b\rangle \cong \Bbb Z_5\times \Bbb Z_3$. It concludes that ${\rm Aut}(H)\cong{\rm Aut}(\Bbb Z_5)\times{\rm Aut}(\Bbb Z_3)$. The reason is explained in the book that "since these two subgroups are characteristic in $H$."
From my understanding, the explanation in the book implies that if $G=A\times B$ and $A$ and $B$ are characteristics in $G$, then ${\rm Aut}(G)={\rm Aut}(A)\times{\rm Aut}(B)$.
Is this true statement?
What I don't understand is how the characteristics of two subgroups matters.
I am a little baffled because there is no section in the book that talks about this. Can you please give me some insight?