Local subgroup of finite group is contained in p-local subgroup for some prime p

Statement
Suppose $$G$$ is a finite group and $$H$$ is a local subgroup of $$G$$, i.e., it is the normalizer of some nontrivial solvable subgroup of $$G$$. Then, there exists a prime number $$p$$ and a p-local subgroup $$K$$ of $$G$$ such that $$H \le K$$.

Proof idea
Suppose $$H = N_G(Q)$$ for a solvable subgroup $$Q$$ of $$G$$. The idea is to find a prime $$p$$ such that $$O_p(Q)$$ (the p-core of $$Q$$) is nontrivial. Let $$K = N_G(O_p(Q))$$. We show that $$H \le K$$, completing the proof.