No common composition factor with quotient group not implies complemented

Statement
It is possible to have a finite group $$G$$ having a normal subgroup $$N$$ such that $$N$$ and $$G/N$$ have no common composition factors, but $$N$$ is not a permutably complemented subgroup of $$G$$.

Similar facts

 * No nontrivial homomorphism to quotient group not implies complemented

Opposite facts

 * Normal Hall implies permutably complemented

Proof
Let $$G = SL(2,5)$$, the special linear group of $$2 \times 2$$ matrices over the field of five elements. Let $$N = Z(G)$$. Then, $$N$$ is a subgroup of order two and $$G/N$$ is isomorphic to the alternating group of degree five, which is simple. Thus, $$N$$ and $$G/N$$ have no common composition factors. However, $$N$$ has no complement in $$G$$.