That is: is it true that if projective k[G]-modules have same composition factors then they are isomorphic?

This is easy to see for char(k)=0, or if G is a composition of a p-group and a p′-group. Serre in "Linear Representations of Finite Groups" (a remark in 16.2 after Corr.2) states this as a well-known fact: "Indeed we know that the equality [P]=[P′] ... is equivalent to P=P′". But unfortunately no references.