Composition factor-equivalent groups

Definition
Suppose $$G$$ and $$H$$ are groups of finite composition length. We say that $$G$$ and $$H$$ are composition factor-equivalent if the multisets of composition factors of $$G$$ and $$H$$ are the same (possibly, occurring in a different order). In other words, every defining ingredient::simple group has the same number of isomorphic copies in the defining ingredient::composition series of $$G$$ as in the composition series of $$H$$.

Facts

 * The trivial group is not composition factor-equivalent to any other group.
 * A simple group is not composition factor-equivalent to any other group.
 * Any two finite groups that are composition factor-equivalent must have the same order.
 * Two finite solvable groups are composition factor-equivalent if and only if they have the same order.