I was playing around with semidirect products and tried finding a non abelian semi direct product of $\mathbb{Z}_2\times \mathbb{Z}_2\rtimes \mathbb{Z}_2$. I couldn't find a group that worked, and I realized this was because the non abelian groups of order 8 ($D_8$ and $Q$) didn't have $\mathbb{Z}_2\times\mathbb{Z}_2$ as a subgroup.
So my question is whether we can identify when a group can be expressed as a semi-direct product that includes $\mathbb{Z}_2\times\mathbb{Z}_2$. For example, how do we know if more than one group of order 20 can be written as a semidirect product of $\mathbb{Z}_2\times\mathbb{Z}_2$?
(I say "more than one" because of the direct product $\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_5$, which is technically a semidirect product)
Thanks a lot.
But $D_8$ does have $\mathbb{Z}_2 \times \mathbb{Z}_2$ as a subgroup: using the presentation $$D_8 = \langle a,b \mid a^2 = b^2 = (ab)^4 = Id\rangle$$ the subgroup generated by $a$ and $bab$ is isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_2$.
To get a non-abelian semidirect product $(\mathbb{Z}_2 \times \mathbb{Z}_2) \rtimes \mathbb{Z}_2$ all you need is an order 2 isomorphism of $\mathbb{Z}_2 \times \mathbb{Z}_2$, for example swap the two generators.
But there are no nonabelian groups of order 20 which are semidirect products of the form $$(\mathbb{Z}_2 \times \mathbb{Z}_2) \rtimes \text{(some order 5 group)} $$ because the only order 5 group is cyclic and there are no order 5 automorphisms of $\mathbb{Z}_2 \times \mathbb{Z}_2$.