Let $k$ be a field and $Q$ be the quiver with two vertices 1 and 2 and three arrows: $a$ from 2 to 1, $b$ from 2 to 1 and $c$ from 2 to 2. Let $I_1=\langle ab-c^2,ba\rangle$ and $I_2=\langle ab-c^2,c^4,ba-bca\rangle$ and $A_1:=kQ/I_1$ and $A_2:=kQ/I_2$ be the corresponding quiver algebras. It is known that $A_1$ and $A_2$ are isomorphic if and only if the characteristic of the field is not equal to two. Let $M_1$ be the direct sum of all indecomposable $A_1$ modules and $M_2$ be the direct sum of all indecomposable $A_2$ modules. Let $B_1=\operatorname{End}_{A_1}(M_1)$ and $B_2=\operatorname{End}_{A_2}(M_2)$ be the Auslander algebras of $A_1$ and $A_2$. Let $C_1=\underline{\operatorname{End}_{A_1}}(M_1)$ and $C_2=\underline{\operatorname{End}_{A_2}}(M_2)$ be the stable Auslander algebra of $A_1$ and $A_2$.

Question 1: Exercise 2 (e) in chapter VII. in the book on Artin algebras by Auslander, Reiten and Smalo asks for a prove that when the characteristic of the field $k$ is equal to two, $A_1$ and $A_2$ have the same Auslander–Reiten quiver (this follows because they are socle-equivalent), but $B_1$ and $B_2$ are not isomorphic. Is there an easy prove/argument for this using only the techniques developed in the book? A direct proof calculating quiver and relations of $B_1$ and $B_2$ would probably be very tedious (both algebras have 20 indecomposable modules) and longer than 10 pages. But maybe there is an elementary argument.

Question 2: Are $B_1$ and $B_2$ derived or stable equivalent in characteristic 2?

Question 3: Are $C_1$ and $C_2$ isomorphic?