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.

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 answers on Questions 1 and 2.

Y.B.

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.

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.