  In this paper we classify, up to equivalence, all semisimple nontrivial Hopf
algebras of dimension $2^{2n+1}$ for $n\geq 2$ over an algebraically closed
field of characteristic $0$ with the group of group-like elements isomorphic to
$\mathbb{Z}_{2^{n}}\times \mathbb{Z}_{2^{n}}$. Moreover we classify all such
nonisomorphic Hopf algebras of dimension $32$ and show that they are not
twist-equivalent to each other. More generally, given an abelian group of order
$2^{m-1}$ we give an upper bound for the number of nonisomorphic nontrivial
Hopf algebras of dimension $2^{m}$ which have this group as their group of
group-like elements.
