Trivial endomorphisms of the Calkin algebra

We prove that it is consistent with ZFC that every unital endomorphism of the Calkin algebra Q(H) is unitarily equivalent to an endomorphism of Q(H) which is liftable to a unital endomorphism of ℬ(H). We use this result to classify all unital endomorphisms of Q(H) up to unitary equivalence by the Fredholm index of the image of the unilateral shift. As a further application, we show that it is consistent with ZFC that the class of C*-algebras that embed into Q(H) is not closed under tensor product nor countable inductive limit.