Order has only two prime factors implies prime divisor with larger class two subgroups is core-nontrivial

Statement
Suppose $$p$$ and $$q$$ are two distinct primes, and $$G$$ is a group of order $$p^aq^b$$ for nonnegative integers $$a,b$$. Suppose $$e_p(G)$$ denotes the maximum of the orders of $$p$$-subgroups of $$G$$ of nilpotence class two. Then, if $$e_p(G) > e_q(G)$$, $$p$$ is a core-nontrivial prime divisor of $$G$$: $$G$$ has a normal $$p$$-subgroup.

Related facts

 * Order has only two prime factors implies solvable
 * order has only two prime factors implies prime divisor with larger prime power is core-nontrivial except in finitely many cases