Order statistics-equivalent finite groups

Definition
Two finite groups $$G$$ and $$H$$ are termed order statistics-equivalent if they satisfy the following equivalent conditions:


 * 1) $$G$$ and $$H$$ have the same order statistics.
 * 2) There is a bijection from $$G$$ to $$H$$ that sends any element to an element of the same order.
 * 3) There is a finite group $$K$$ containing a subgroup $$G_1$$ isomorphic to $$G$$ and a subgroup $$H_1$$ isomorphic to $$H$$ such that, for any conjugacy class in $$K$$, its intersection with $$G_1$$ has the same size as its intersection with $$H_1$$.
 * 4) There is a finite group $$K$$ containing a subgroup $$G_1$$ isomorphic to $$G$$ and a subgroup $$H_1$$ isomorphic to $$H$$ such that the induced representation of $$K$$ from the regular representation of $$G_1$$ is equivalent as a linear representation to the induced representation of $$K$$ from the regular representation of $$H_1$$.

Note that any two order statistics-equivalent finite groups have the same order.

Relation with group properties
Note that the third column (first question column) is the conjunction of the fourth and fifth columns.