This is a somewhat broad question. I would like to get an idea of what is known about mapping class groups (MCG) of a Seifert manifold $M$, in particular how to systematically compute it.

I want to restrict to Haken ones for now, and it is known that $\text{MCG}(M)$ is isomorphic to $\text{Out}(\pi_1(M))$ (c.f. page 33 of https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/3-manifold-groups-final-version-031115).

My main focus is $M$ over an $S^2$ base, namely $S^2\left(b;\frac{a_1}{b_1},\dots,\frac{a_n}{b_n}\right)$ (sorry for this non-standard notation). A potentially useful reference is https://arxiv.org/abs/math/0010077 by D. McCullough, whose Table 4 contains only $S^2(2,2,m), S^2(2,3,3), S^2(2,3,4), S^2(2,2,5)$. Their MCGs are known according to Section 3 in the same paper.

I would like to know at least the MCGs of Seifert manifolds $S^2\left(\frac{a_1}{b_1},\frac{a_2}{b_2},\frac{a_3}{b_3}\right)$, which are beyond the scope of the paper by McCullough.