Normal Subgroups of Solvable Groups with elementary abelian Sylow 2 subgroup

Question

I am trying to understand implications of the composition series for groups with elementary abelian Sylow 2 subgroups, to see how much of the odd part of the group can be separated. Suppose G has order $2^k m$, where m is odd. If k is 1, then the Sylow 2 subgroup has a normal complement which has odd order, i.e., the group has a subgroup of index 2. If k = 2, the Sylow subgroup has order 4. All my tests with GAP find that such a group has either a normal subgroup of index 4, or a normal subgroup of order 4. Is it true that every solvable group of order 4m, with m odd has either a normal subgroup of index 4, or a normal subgroup if index 4? More generally, is it true that for a solvable group of order $2^k m$, with m odd, and elementary abelian Sylow 2 subgroup has either a normal subgroup of order $2^k$, or a normal subgroup of index $2^k$?

Answer 1

No. Try $\mathtt{SmallGroup}(324,160)$ in GAP.