Let $G$ be a group of order $2^n$. Does $G$ have a normal abelian subgroup of order at least $2^{n/2}$?

(This is true, via computations in GAP, for $n \le 8$. The question is similar to one posed here: https://math.stackexchange.com/questions/44275/abelian-subgroups-of-p-groups/44283#44283 However, that question, and answer, involves groups of order $p^n$, for odd primes $p$, and I need $p$ to be even!)