There exists a 2-group with a maximal elementary abelian subgroup of order four and rank four

Statement
There exists a finite $$2$$-group $$P$$ having a maximal elementary abelian subgroup $$A$$ of order four (i.e., a Klein four-group that is not contained in any bigger elementary abelian subgroup), and such that $$P$$ has an elementary abelian subgroup $$B$$ of rank $$4$$ (i.e., of order $$16$$). In other words, for the prime two, there can be a fact about::group of prime power order having a maximal elementary abelian subgroup of prime-square order that has rank four.

Related facts

 * Maximal elementary abelian subgroup of prime-square order implies normal rank at most the prime
 * Maximal elementary abelian subgroup of order four implies subgroup rank at most four
 * Maximal elementary abelian subgroup of prime-square order implies rank at most the prime for odd prime