I've just proven that every Group has an abelian Subgroup $A$ such that $A < B \leq G \rightarrow B$ is nonabelian using Zorn's Lemma. Heres a sketch of said proof: Let $\Omega$ be the set of abelian subgroups of G(this set is obviously partially ordered by inclusion and nonempty as it contains the trivial subgroup) and $I_0\subseteq I_1 \subseteq ...$ a chain in $\Omega$. Then $\bigcup_{i \in \mathbb{N}} I_i$ is a supremum of said chain in $\Omega$. By Zorn's Lemma $\Omega$ must contain maximal Elements.
Now I'm trying to characterize Groups that have only one such maximal subgroup. But I don't really see the property of a Group that makes it have only a single such subgroup.
Only abelian groups have one maximal abelian subgroup.
Assume that $H$ is a maximal abelian subgroup of a nonabelian group $G$. Pick any $x\not\in H$ (which exists since $H\neq G$). Then $\langle x\rangle$ is abelian and thus contained in some maximal abelian subgroup $H'$. Obviously $H$ and $H'$ cannot be equal.