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

Statement
Suppose $$k$$ and $$c$$ are natural numbers and $$0 \le k \le 4$$. Suppose $$p$$ is a prime number. Then, if $$G,H$$ are groups both of order $$p^k$$ and nilpotency class exactly $$c$$, $$G$$ and $$H$$ must be fact about::commutator map-equivalent groups.

Corollaries

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