TITLE: A simple question on characteristic subgroups
QUESTION [5 upvotes]: Suppose $P$ is a finite abelian $2$-group and $U$,$V$ are characteristic subgroups of $P$ such that $|V:U|=2$. Does it follow that $P$ has a characteristic subgroup of order $2$?

REPLY [4 votes]: To answer a (deleted now, I guess) question of Stefanos in the comments: show a finite simple group with a Sylow 2-subgroup satisfying the hypothesis on P has order 2. One uses transfer and coprime action.
Proposition: Suppose G is a finite group with abelian Sylow 2-subgroup P such that U ≤ V ≤ P are characteristic in P with [V : U] = 2.  Then G has a normal subgroup of index 2.
Proof: Let N = NG(P) and since P is abelian, let A = N/P be the induced group of automorphisms. Since the action of A on P is coprime, V = CV(N) × [ V, N ], but since U is normal in N, [ V, N ] ≤ U < V is a proper subgroup of V, and so CV(N) ≠ 1.  Of course, then CP(A) ≥ CV(N) ≠ 1 and [ P, N ] is a proper subgroup of P.  [ P, N ] is known as the focal subgroup of P in G (because P is abelian), and G has a quotient isomorphic to the (non-identity) 2-group P/[ P, N ], and so G has a normal subgroup of index 2 corresponding to a maximal subgroup of P under the transfer map. $\square$.
The key part of the argument is that [ V, N ] ≠ V.  In more detail: let v be an element of V not contained in U.  Since U is normal in N, no conjugate of v can be in U, and so N permutes the non-identity elements of V/U by conjugation.  In particular, (vU)n must be a non-identity coset of V/U, but there is only one: vU.  Hence [ v, n ] = u, and [ V, N ] ≤ U.
I use that P is abelian, that U, V are normal in N, and that [ V : U ] = 2, not just that it is prime.
Indeed, G non-abelian of order 6 satisfies similar hypotheses when p=3, with 1 = U ≤ V = P, but G has no normal subgroup of index 3 (the corresponding conclusion). It is also necessary to assume P abelian in order to get CV(N) ∩ [ V, N ] = 1.  For example, P ≅ Q8 has 1 = U ≤ V = Z(P) ≤ P characteristic, and NG(P) always controls G-fusion (and so transfer), but G = SL(2, q) for q ≡ ±3 mod 8 has P as a Sylow 2-subgroup and no normal subgroup of index 2.