This is a set of observations that should help you understand your inductions of non-trivial characters. If I read Marty's answer correctly, this should be a concrete way of understanding what he is saying using your particular example.

First note that each non-trivial character $\chi$ has a conjugate character $\overline{\chi}$ and that the inductions of $\chi$ and $\overline{\chi}$ must also be conjugate.  Next, irreducible characters occur in sets which are related to one another by outer automorphisms of the dual group, these sets are defined by those irreducible characters with common kernel. Finally, note that the induction of the regular representation of the subgroup is the regular representation of the big group.

How does this help? Well in your example, lets look at what remains from the regular representation of $GL_3(\mathbb{F}_2)$ after we take away the induction of the trivial character of the Sylow 7-group. The remaining pieces are:

$3V_{3a} \oplus 3V_{3b} \oplus 6V_6 \oplus 6V_7 \oplus 6V_8$

Each induction of the 6 nontrivial characters must have dimension 24, and since the nontrivial characters are all more or less equivalent (up to outer automorphisms of the dual group to be technical), all their inductions should look more or less the same as well. This means that each induces to either $W = V_{3a}\oplus V_6 \oplus V_7\oplus V_8$ or its conjugate $\overline{W} = V_{3b}\oplus V_6 \oplus V_7\oplus V_8$.

With a bit of extra work, you would then find that if a character $\chi$ of the Sylow 7-subgroup induces to $W$ then so do the characters of $\chi^{\otimes 2}$ and $\chi^{\otimes 4}$ while the characters of $\chi^{\otimes 3}$, $\chi^{\otimes 5}$, and $\chi^{\otimes 6}$ will induce to $\overline{W}$, although the particular split on the exponents don't matter to your particular question (as currently stated).