Normal not implies subgroup-conjugating

Statement
A normal automorphism of a group (an automorphism that preserves all the normal subgroups) need not be subgroup-conjugating -- it need not send every subgroup to a conjugate subgroup.

Stronger facts

 * Class-preserving not implies subgroup-conjugating: An automorphism of a group that preserves conjugacy classes of elements need not send every subgroup to a conjugate subgroup.

Facts used

 * 1) Class-preserving not implies subgroup-conjugating
 * 2) Class-preserving implies normal

Proof
The proof follows directly by piecing together facts (1) and (2).