Nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order

Statement
Suppose $$p$$ is a prime number and $$0 \le k \le 4$$. Then, for a group of order $$p^k$$, the fact about::nilpotency class $$c$$ of the group determines its conjugacy class size statistics.

Here is the complete list:

Stronger facts

 * Nilpotency class and order determine group up to commutator map-equivalence for up to prime-fourth order

Other similar facts

 * Nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order

Opposite facts

 * Nilpotency class and order need not determine conjugacy class size statistics for groups of prime-fifth order