Let $\Sigma(p,q,r)$ be the Brieskorn homology 3-sphere with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$ (so not the 3-sphere or the Poincare sphere). The fundamental group is given by $$ \pi_1(\Sigma(p,q,r)=\langle a,b,c\, |\, a^p=b^q=c^r=abc \rangle $$, the centrally extended triangle group for $(p,q,r)$. The isotopy group $\pi_0(Diff(\Sigma(p,q,r)))$ must be finite as well as the outer automorphisms of the fundamental group $Out(\pi_1(\Sigma(p,q,r))$. Furthermore the map $$\pi_0(Diff(\Sigma(p,q,r)))\to Out(\pi_1(\Sigma(p,q,r))$$ is injective ( I found this result in D. McCullough, Virtually geometrically finite mapping class groups of 3-manifolds, J. Differential Geom. 33 (1991), no. 1, 1–65.) Now to my questions:

- Are there new results to show that this map is an isomorphism? Or, what is the order of the isotopy group?
- In particular, I'm interested in the example $\Sigma(2,5,7)$ (bounding a contractable smooth 4-manifold).
- I do not find any result about the outer automorphism of tre centrally extended triangle group. What is known?