Full automorphism groups of the axial algebra for $M_{11}$ and related algebras

In this paper, in continuation of arXiv:2311.18538, we compute the full automorphism groups of the 286-dimensional algebra for $M_{11}$, its subalgebras and other related algebras. This includes, in particular, the 101-dimensional algebra for $L_2(11)$ and the 76-dimensional algebra for $A_6$. While smaller algebras can be handled by the fully automatic nuanced method from arXiv:2311.18538, the larger algebras, mentioned above, require a hybrid method combining computation with hand-made proofs.