I am using Dummit and Foote (pg 181-182) and trying to understand section 5.5 on semidirect product for a group of order 30. They start out with a reminder that if $H$ is of order 15 it is a normal subgroup (because of index 2) as well as cyclic, thus we can form a group $G=HK$ where $H\cap K = 1$ then $G\cong H\rtimes K$ for some $\phi:K\rightarrow Aut(Z_{15})$ where $Aut(Z_{15})\cong(\mathbb Z/15 \mathbb Z)^{\times}\cong Z_4\times Z_2$ (I understand how they got this).
Then the book explains that the given $Aut(H)$ has three elements of order 2, one of those elements whose actions on $H = \langle a\rangle\times \langle b\rangle$ is $\{a\mapsto a, b\mapsto b^{-1}\}$ (there are two others the book shows but I just want to understand one of first). If a given $K=\langle k\rangle$ with the action illustrated above the book says that $G = H\rtimes_{\phi_1}K\cong Z_5\times D_6$. My questions are the following:
1) I am not completely sure why $Aut(H)$ has three elements of order $2$ precisely.
2) I do not understand the final conclusion, where did the dihedral group come from and where did $Z_5$ come from?
I think if I understand this, the other two non-isomorphic groups should make more sense. Thanks!