On the $\mathfrak F$-residual of the direct product of finite groups

Let $\pi$ be a subset of the set $\mathbb P$ of all primes, and let $\pi'=\mathbb P\backslash\pi$. A formation $\mathfrak F$ is called $\pi'$-saturated if $G/O_{\pi'}(\Phi(G))\in\mathfrak F$ implies $G\in\mathfrak F$. If $\mathfrak F$ is a nonempty $\pi'$-saturated formation of $\pi$-soluble groups, then it is proved that $(A\otimes B)^\mathfrak F=A^\mathfrak F\otimes B^\mathfrak F$ for any finite groups $A$ and $B$. In the case $\pi=\mathbb P$, this result was proved by K. Doerk and T. Hawkes in 1978.