$\DeclareMathOperator{\mod}{\mathrm{mod}}$When I’m learning the $\textit{representation theory of Artin algebras}$ wrote by M. Aulander, I. Reiten and S.O. Small, I find two questions in chapter X.

Firstly, if $F:\mod \Lambda\to \mod\Lambda’$ is an exact functor, and induces a stable equivalence, then $F$ takes an almost split sequence in $\mod \Lambda$ to almost split sequence in $\mod\Lambda’$, where $\Lambda$ and $\Lambda’$ are Artin algebras.

Secondly, if $\Lambda$ is self-injective, then $\Omega$ induces a stable equivalence and we also have a method to get a new almost split sequence from an almost split sequence.

I cannot prove above, but if I can prove the following claim, then I can prove the above:

If $\Lambda$ is an Artin algebra, $A$ and $C$ are indecomposable modules, $B$ doesn't have projective direct summand, $P$ is a projective module, $$\delta:0\to A\to B\coprod P\to C\to 0$$ is an exact sequence. If we have another exact sequence $$0\to A\to B\coprod Q\to C\to 0,$$ with $Q$ is a projective module, then $Q\simeq P$ as $\Lambda$ modules.

If the above claim is not true, is it true for $\delta$ is an almost split sequence?