Inductive upper bound on number of groups of prime power order using power-commutator presentations

Statement
Suppose $$p$$ is a prime number and $$n$$ is a natural number. We have the following very crude inductive upper bound on the number of isomorphism classes of groups of order $$p^n$$ (denoted $$f(p,n)$$) in terms of the number of isomorphism classes of groups of order $$p^{n-1}$$ (denoted $$f(p,n-1)$$):

$$\! f(p,n) \le p^{n(n-1)/2}f(p,n-1)$$

Related facts

 * Upper bound on number of groups of prime power order using power-commutator presentations
 * Higman-Sims asymptotic formula on number of groups of prime power order