The trivial intersection problem for characters of principal indecomposable modules

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For a prime p and a finite group G let Φ p ( G ) denote the complex character associated to the projective indecomposable module in characteristic p with trivial head. Let Irr ( Φ p ( G ) ) denote the set of irreducible characters occurring as constituents in Φ p ( G ) . We characterize all finite simple groups which satisfy Irr ( Φ p ( G ) ) ∩ Irr ( Φ q ( G ) ) = { 1 G } for all primes p ≠ q .