A group $ Q $ is called quasisimple if it is a perfect central extension of a simple group.
A group $ A $ is called almost simple if there exists a series of containments $ S \subset A \subset Aut(S) $ where $ S $ is a simple group and $ Aut(S) $ is the automorphism group of $ S $. In other words, a group $ A $ is almost simple if it is the extension of a simple group by some subgroup of the outer automorphism group.
For a quasisimple group $ Q $, all characters of the corresponding simple group $ Q/Z(Q) $ lift to characters of $ Q $. Since the characters of finite simple groups are well known, these are easy to obtain. There are some other characters of $ Q $ which cannot be obtained from the characters of the simple group $ Q/Z(Q) $. Even these are not too hard to find since quasisimple groups are perfect and there are good libraries of characters for (small) perfect groups. For example GAP has the characters tables of all Perfect Groups of size up to two million. https://www.gap-system.org/Datalib/perfect.html
The characters of finite almost simple groups are also not too hard to obtain. They can usually be found in the same references as the characters tables of finite simple groups. In GAP one can just use the command AutomorphismGroup(S) where $ S $ is the simple group in question. For a given almost simple group $ A $ most (but not all) of its characters can be found by just restricting the characters of $ Aut(S) $ to the subgroup $ A $.
For an almost quasisimple group $ G $ I am unsure how to find characters. Some characters can be lifted from the character table of $ Aut(G) $. However I think there are many that cannot. For example there is a group of size $ 1440 $ which is an extension of $ SL(2,9) $ by an outer automoprhism of order $ 2 $ and it appears to have a degree $ 4 $ character. However I do not know how to compute this character.
Even easier than computing the full character table, I would just be curious how to find all the irrep dimensions for an almost quasisimple group.
TLDR: I'm curious how to find the character tables of almost quasisimple groups. Even figuring out all the irrep dimensions for (small) almost quasisimple groups would be interesting to me. I would be happy with a reference for the character tables, or a way to find them in GAP, or some direct method to find irrep dimensions for almost quasisimple groups, as long is works for a few interesting small groups, say of size less than 10,000.
Edit 1: very interesting point raised in the comments by Derek Holt. In response here is a (perhaps slightly vague but hopefully no longer incorrect) definition: A group $ G $ is called almost quasisimple if it is the extension of a quasisimple group by some subgroup of its outer automorphism group, with the outer automorphisms acting in the natural way.