Composition factor-unique not implies composition series-unique

Statement
It is possible to have a group (in fact, a finite group) such that any two composition series for the group have the same composition factors occurring in the same order, but such that there is more than one composition series.

Related facts

 * Finite not implies composition factor-unique
 * Finite not implies composition factor-permutable

Direct product of isomorphic simple groups
Suppose $$S$$ is a simple group. Let $$G := S \times S$$ be the direct product of isomorphic copies of $$S$$. Note that we have two composition series:

$$\{ e \} \le S \times \{ e \} \le S \times S$$

and

$$\{ e \} \le \{ e \} \times S \le S \times S$$.

Thus, $$G$$ does not have a unique composition series. On the other hand, any composition series of $$G$$ has the simple group $$S$$ occurring twice, so all composition series have the composition factors occurring in the same order.

(Note that when $$S$$ is simple non-Abelian, these are the only two composition series: this follows from the fact that normal subdirect product of perfect groups equals direct product). When $$S$$ is simple Abelian, it is a cyclic group of prime order $$p$$, and there are $$p + 1$$ possible composition series.