Automorphism group of a semi-direct product

Question

I'm trying to construct the semi-direct product $(\mathbb{Z}_7 \rtimes \mathbb{Z}_3) \rtimes \mathbb{Z}_2$. Constructing the first factor in parentheses is not difficult. But when it comes to constructing the entire semi-direct product I'm having issues finding the maps $\phi: \mathbb{Z}_2 \rightarrow \text{Aut}(\mathbb{Z}_7 \rtimes \mathbb{Z}_3)$. First, with regard to the semi-direct product which I mentioned in the beginning, does such a thing actually exist, or should it be $(\mathbb{Z}_7 \times \mathbb{Z}_3) \rtimes \mathbb{Z}_2$ or $(\mathbb{Z}_7 \rtimes \mathbb{Z}_3) \times \mathbb{Z}_2$? Also, I'm not so familiar with automorphism groups of semi-direct products, which I believe is the reason I'm not getting very far. I would appreciate some help here.