Let $A$ be a finite abelian group, and consider the wreath product $A\wr \mathbb{Z}/2= (A\oplus A)\rtimes\mathbb{Z}/2$.
Is it possible to describe all the complements $K$ of the subgroup $A\oplus 0\subseteq A\wr \mathbb{Z}/2$?
I am particularly interested in understanding when $K$ is in addition assumed to be normal!
If $A$ has odd order, I believe that the only such complement is $\Delta\rtimes\mathbb{Z}/2$ where $\Delta = \{(a,a)| a\in A\}\subseteq A\oplus A$. The situation seems to be much more complexe in the presence of 2-torsion.