Concerning the finite general linear groups (or other finite groups of Lie type), it's been known for a long time that you can't get every irreducible character as a constituent of one induced from the trivial character of a proper nontrivialparabolic subgroup. This is what makes the whole subject so challenging, going back as far as the work of Frobenius for the groups $\mathrm{SL}_2(\mathbb{F}_p)$ and extending through the work of J.A. Green on characters of finite general linears to the much more complicated work coming from Deligne-Lusztig theory. Caveat: In this situation I'm not considering all possible proper subgroups, just those relevant to the BN-pair structure, so it's of course possible to find exceptions. But in Lie theory, including finite general linear groups, one is really looking for uniform methods to produce character tables.
On the other hand, for groups of Lie type there is a rich theory of what can be done if you induce up from the trivial character of a parabolic subgroup and then decompose the induced character using Hecke algebra methods. The problem is that it doesn't get everything you want.
By the way, I'd be curious to know whether there is a reasonable necessary and sufficient condition on a finite group to make the answer to your question affirmative. (Probably not.) In any case, your header does suggest that you want the induced representations involved to be irreducible, which misleads people at first.