Order statistics-equivalent finite groups

This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

Definition

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

$G$ and $H$ have the same order statistics.
There is a bijection from $G$ to $H$ that sends any element to an element of the same order.
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 $H_{1}$ .
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.

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

Group property	Finite version	Is any group order statistics-equivalent to a group with this property isomorphic to it?	Does any group order statistics-equivalent to a group with this property also have this property?	Are any two groups with this property that are order statistics-equivalent also isomorphic?
Cyclic group	Finite cyclic group	Yes: Finite group having the same order statistics as a cyclic group is cyclic	Yes	Yes
Abelian group	Finite abelian group	No	No; for instance: Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring	Yes: Finite abelian groups with the same order statistics are isomorphic
Nilpotent group	Finite nilpotent group	No	Yes: Order statistics of a finite group determine whether it is nilpotent	No; for instance: Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring

Relation	Meaning	Proof of implication	Proof of strictness (reverse implication failure)
1-isomorphic finite groups (finite case)	there is a bijection between the groups that is a 1-homomorphism of groups and whose inverse is also a 1-homomorphism of groups	1-isomorphic implies order statistics-equivalent	order statistics-equivalent not implies 1-isomorphic
Order-cum-power statistics-equivalent finite groups
Root statistics-equivalent finite groups