For a finite group $G$ and field $k$ of char$=p$, if $P,P'$ are projective $k[G]$-modules with $[P]=[P']$, is it true that $P=P'$?

Question

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 $\text{char}(k)=0$, 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.