Finite group generated by Schur-trivial subgroups of relatively prime indices is Schur-trivial

Statement
Suppose $$G$$ is a finite group. Suppose $$H_1,H_2,\dots,H_n$$ are subgroups of $$G$$ such that the values of the indices $$[G:H_i], 1 \le i \le n$$, have gcd 1 (note that we do not assume that any two of them are relatively prime -- we only assume that there is no prime factor common to all the indices). Suppose further, that each $$H_i$$ is a Schur-trivial group. Then, the group $$G$$ is also a Schur-trivial group.

Related facts

 * Cyclic implies Schur-trivial
 * All Sylow subgroups are Schur-trivial implies Schur-trivial