Definition A $2$-group $S$ is called wreathed if it is isomorphic to $(C\times C)\rtimes \langle i \rangle$ where $C$ is a cyclic group of order $2^n$ and $i$ is an involution with action $(a,b)^i=(b,a)$ for all $(a,b)\in C\times C$.
I wonder about the structure of maximal subgroups of $S$. Clearly, one of them is $C\times C$ and it is abelian.
I think the rest of the maximal subgroups is not abelian when $n\geq 2$. (When $n=1$ we have a dihedral group of order $8$.) However, if $M$ is a maximal subgroup of $S$ then $M$ has an abelian subgroup of index $2$.
Any remark and reference is welcome.