Converse of congruence condition for prime power Sylow numbers

Statement
Suppose $$p,q$$ are distinct primes and $$r$$ is a natural number such that:

$$q^r \equiv 1 \mod p$$.

Then, there exists a finite group in which the number of $$p$$-Sylow subgroups (i.e., the $$p$$-Sylow number) equals $$q^r$$.

Related facts

 * Converse of congruence condition for Sylow numbers for the prime two
 * Congruence condition on Sylow numbers

Proof idea
The idea is to take the semidirect product of the additive group of the field with $$q^r$$ elements by a cyclic subgroup of order $$p$$ in its multiplicative group.