homological algebra – flat module and projective module

In Rotman's book "Introduction to Homological Algebra", Theorem 3.62
Let $ 0 rightarrow K rightarrow F rightarrow A rightarrow 0 $ to be an exact sequence of good R-modules, where $ F $ is free. What follows is equivalent
begin {align}
& (1) A is flat \
& (2) For each v in K, there is a R maps theta: F rightarrow K with theta (v) = v.
end {align}

My problem is that this is not the second condition implies that $ A $ is a direct summary of the free module $ F $, therefore projective. So does not this theorem say that flat implies projective? And we already know that projective modules are flat. So, this theorem does not say that projective is equivalent to flat, which is not true in general. for example $ mathbb Q $ like a $ mathbb Z $-module. So, where am I wrong?