Tessellate S.T.E.M.S - Mathematics - College - Set 1 - Problem 1

Suppose $G$ is a non-abelian group, $H$ a normal subgroup of $G$ contained inside the centralizer of $G$ . Suppose $G/H$ is abelian. Then which of the following optiions are correct?

1. if any non-trivial, proper subgroup of the group of automorphisms of $G$ is abelian, it is cyclic.
2. if the group of automorphisms of $G$ is non-abelian, it contains no subgroup which is non-trivial, proper and normal.
3. if the group of automorphisms of $G$ is non-abelian, it contains no subgroup which is non-trivial, proper, normal and abelian.
4. if the group of automorphisms of $G$ is non-abelian, it contains a subgroup which is non-trivial, proper, normal and abelian.

This problem is a part of Tessellate S.T.E.M.S.