On Core-2 Groups

A group G is core-2 if and only if H/HG ≤ 2 for every H ≤ G. We prove that every core-2 nilpotent 2-group of class 2 has an abelian subgroup of index at most 4. This bound is the best possible. As a consequence, every 2-group satisfying the property core-2 has an abelian subgroup of index at most 16.