Embedding of $\mathbb{Z}$

Question

In a certain exercise I am asked to prove something that involves semidirect product. I wanted to know if $\mathbb{Z}$ could be embedded in a semidirect product of bigger groups, named for example $A,B$, this is, if there exists, $A,B$ so that $\mathbb{Z} < A\rtimes_{\theta} B$ for some $\theta: B \rightarrow Aut(A)$. Is it possible to simply assume this?

Answer 1

Any direct product is a semidirect product, so just take

$$\Bbb Z(\cong\circ \le)\Bbb Z\times \Bbb Z_2,$$

where $(\cong\circ \le)$ means "isomorphic to a subgroup of".

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Less glib, look at the semidirect products of $\Bbb Z$ with itself, discussed here.