A condition for a finite group to be a Schmidt group

Let $G$ be a finite group $G$, and let $\pi$ be a set of primes such that $2\in \pi$. We prove that if all maximal subgroups of $G$ are $\pi$-closed and $G$ itself is not $\pi$-closed then $G$ is a Schmidt group. The proof employs the author's earlier results on the properties of pairs $(G,\pi)$ where $G$ is a simple minimal non-$\pi$-closed group and $\pi$ is arbitrary.