The Sylow subgroup $\Sigma_n$ of the Symmetric group $S_{p^n}$ has only one maximal normal abelian subgroup, say $B$, for $p>2$ and $n>1$. As $\Sigma_n/B$ is isomorphic to
$\Sigma_{n-1}$, the group $\Sigma_n$ yields a positive answer on Question 2.
Y.B.
