Normal subgroup of rational group implies any two elements generating the same cyclic subgroup are automorphic

Statement
Suppose $$G$$ is a fact about::rational group and $$N$$ is a normal subgroup of $$G$$. Then, $$N$$ is a fact about::group in which any two elements generating the same cyclic subgroup are automorphic.

Related facts

 * Alternating group implies any two elements generating the same cyclic subgroup are automorphic
 * Normal subgroup of ambivalent group implies every element is automorphic to its inverse