Let $ S $ be a finite (non-abelian) simple group. Then there always exists a natural extension of $ S $ by the outer automorphism group $ Out(S) $ with elements of $ Out(S) $ acting as outer automorphisms. Call this group $ S.Out(S) $.
Let $ Q $ be the Schur cover of a finite (non-abelian) simple group $ S $. Does there always exist a natural extension $ Q.Out(Q) $ where the elements of $ Out(Q) $ act as outer automorphisms on the $ Q $ subgroup?
Note that this is not always the case for a quasisimple group $ Q $. For example $ Out(SL(2,9))\cong C_2 \times C_2 $ but there is no almost quasisimple extension $ SL(2,9).Out(SL(2,9)) $. See the comment from Derek Holt here Characters of almost quasisimple groups
The reason that I expect this to be true when $ Q $ is Schur trivial is that it is true in the case of simple Lie groups that there is always a natural group $ S.Out(S) $ if $ S $ is a simply connected simple Lie group, and the Schur cover of a finite (non-abelian) simple group is analogous to the universal cover of a simple lie group.
Best Answer
Let's work with the example of the simple group $ A_6 $ of order $ 360 $.
The theory of quasisimple groups is nice in the sense that "everything you can think to ask for exists." The Schur multiplier of $ A_6 $ is cyclic $ 6 $. So you can ask for the quasisimple groups $ 2.A_6,3.A_6,6.A_6 $ and they all must exist. Fun fact: $ 2.A_6 $ is better known as $ SL(2,9) $.
The theory of almost simple groups is another situation which is nice again in the sense that "everything you can think to ask for exists." $ Out(A_6) $ is $ 2 \times 2 $. So there are four nontrivial subgroups of $ Out(A_6) $: the whole group $ 2 \times 2 $, and three different groups of order $ 2 $ generated by the three different outer automorphisms of order $ 2 $. Again everything we can think to ask for exists: the whole group $ 2 \times 2 = Out(A_6) $ corresponds to $ Aut(A_6) $, while the three outer automorphisms of order $ 2 $ correspond to $ S_6, PGL(2,9) $ and $ M_{10} $ respectively. Fun fact: $ Aut(A_6) $, and thus also the subgroups $ A_6, S_6, PGL(2,9), M_{10} $, all have nice permutation representations of degree $ 12 $.
The theory of almost quasisimple groups, however, is very subtle and not so nice in the sense that we are not guaranteed that everything we can think to ask for exists. Indeed many things we can think to ask for do not exist. Take the case of $ A_6 $. Of course $ 2.A_6,3.A_6,6.A_6 $ all exist. We can also think to ask for almost quasisimple groups of the form $ 2.S_6, 3.S_6, 6.S_6 $, luckily these all exist. We can also think to ask for almost quasisimple groups of the form $ 2.PGL(2,9), 3.PGL(2,9), 6.PGL(2,9) $ again these all exist. However, we can also think to ask for almost quasisimple groups of the form $ 2.M_{10}, 3.M_{10}, 6.M_{10} $. Of these, $ 3.M_{10} $ exists but the other two do not exist. Finally, we can think to ask for almost quasisimple groups of the form $ 2.Aut(A_6), 3.Aut(A_6), 6.Aut(A_6) $. Again $ 3.Aut(A_6) $ exists but the other two do not exist.
Connecting this back to the original question, the Schur cover of $ A_6 $ is $ 6.A_6 $ so the fact that the extension $$ 6.A_6.Out(A_6)=6.Aut(A_6) $$ does not exist provides a counterexample to the conjecture in the original question.
Source: See "groups" sidebar or "standard generators" section at the top of this page https://brauer.maths.qmul.ac.uk/Atlas/v3/alt/A6/ to see which almost quasisimple groups are possible for $ A_6 $.