In this post, it is discussed how a Brieskorn homology sphere $Sigma(a_1,a_2,a_3)$ with $displaystyle frac{1}{a_1}+ frac{1}{a_2}+ frac{1}{a_3} < 1$ is an aspherical manifold with superperfect fundamental group and non-trivial center. Would anyone know what the outer automorphism groups of their fundamental groups are? I’m looking to do semidirect products with these groups as the kernel group with another group ($mathbb{Z} times mathbb{Z}$) as the quotient group, so I need to know the outer automorphism grousp of these groups.