Normal rank two Sylow subgroup for least prime divisor has normal complement if the prime is odd

Statement
Suppose $$G$$ is a finite group, and $$p$$ is the least prime divisor of the order of $$G$$. Suppose $$P$$ is a $$p$$-Sylow subgroup of $$G$$ and the normal rank of $$P$$ is at most two. Suppose further that $$p$$ is odd. Then, there exists a normal p-complement in $$G$$. In other words, there exists a normal subgroup $$N$$ of $$G$$ that is a permutable complement to $$P$$. Thus, $$G$$ is a fact about::p-nilpotent group.

Related facts

 * Cyclic Sylow subgroup for least prime divisor has normal complement: This holds for odd primes as well as for the prime $$2$$.
 * Cyclic normal Sylow subgroup for least prime divisor is central
 * Normal of least prime order implies central