TITLE: Presentations of mapping class groups in dimension $3$
QUESTION [10 upvotes]: For any closed oriented surface $M$, its mapping class group $MCG(M)$ can be generated by Dehn twists along certain curves on $M$. A presentation for the group $MCG(M)$ was found in [1] and then simplified in [2].
How about in dimension $3$? The first question is that, unlike in dimension $2$ in which the classification of spaces is easy (genus and number of boundaries), in dimension $3$ the classification is complicated. One either uses

Thurston's geometrization
Lickorish and Kirby's presentation using links

Neither of them is easy, so I'd expect it much harder to get presentation for the mapping class group.
Question: Nevertheless, is there any presentation known?
In the first view-point, the closest answer I have seen is this, in which Allen Hatcher claims that the MCG of three-manifolds are essentially known [3]. There, the natural map from MCG to $Out(\pi_1(M))$ is considered. While the kernel of this map is understood in the nonprime case [3. section 2], it is not necessarily onto. Even if it's onto, we still do not have a presentation for MCG.
In the second view-point, I have not heard of any result.
Reference

[1] A presentation for the mapping class group of a closed orientable surface-[Hatcher and Thurston]

[2] A simple presentation for the mapping class group of an orientable surface-[Bronislaw Wajnryb]

[3] Stabilization for mapping class groups of 3-manifolds-[Allen Hatcher and Nathalie wahl]

REPLY [2 votes]: For certain 3-manifolds (irreducible), if you are willing to take finite index subgroups, the description is relatively easy, and has to do with dehn-twists too.
Let me add the following paper by McCullough which was useful to me when considering this question: https://projecteuclid.org/journals/journal-of-differential-geometry/volume-33/issue-1/Virtually-geometrically-finite-mapping-class-groups-of-3-manifolds/10.4310/jdg/1214446029.full
I think this goes in line with the comments by HJRW which are more detailed, but I thought this might be useful.