Classification of groups of order a product of a prime-square and another prime

Statement
Suppose $$p$$ and $$q$$ are distinct prime numbers. This article classifies the groups of order $$p^2q$$.

Case that $$p$$ does not divide $$q - 1$$ and $$q$$ does not divide $$p - 1$$
In this case, $$p^2q$$ is an abelianness-forcing number, i.e., all groups of this order are abelian. The two abelian groups are:

Case that $$p$$ divides $$q - 1$$ but $$p^2$$ does not divide $$q - 1$$ and $$q$$ does not divide $$p^2 - 1$$
In this case, there are four isomorphism classes of groups of order $$p^2q$$, given as follows:

Case that $$p^2$$ divides $$q - 1$$
In this case, there are five isomorphism classes of groups of order $$p^2q$$, given as follows:

Special cases: order 12 and order 18
There are five groups of order 12. See classification of groups of order 12. The unusual example is alternating group:A4.

There are five groups of order 18. See classification of groups of order 18.