Famously, all but finitely many finite simple groups are (cyclic or alternating or) of Lie type. The groups of Lie type have central extensions coming from the simply connected covers of the corresponding algebraic groups (and the alternating groups have double covers). Famously, for all but finitely many (alternating or) Lie type groups, this "Lie theoretic" central extension is the Schur cover (universal central extension). Steinberg and Griess classified all of the exceptional Schur covers over a series of papers.
Let us say that a finite group is subsporadic if it is a subgroup of a Schur cover of a sporadic simple group. I know from experience that many exceptional central extensions are subsporadic. For example, the triple cover $3A_6$ of the alternating group $A_6$ is a subgroup of Mathieu's group $M_{24}$, and Conway's group $\mathrm{Co}_1$ contains (via $3\mathrm{Suz}$) a copy of $3A_7$, lifting to the Schur cover $6A_7 \subset 2\mathrm{Co}_1$. Another example: writing $\Omega_n(p)$ for the simple subquotient of $\mathrm{O}_n(\mathbb{F}_p)$, we have $3\Omega_7(3) \subset 3\mathrm{Fi}_{22}$.
Are all exceptional Schur covers subsporadic?
A slight variation of the question is the following. Suppose $G$ is a simple group of Lie type with defining characteristic $p$, and write $n = \mathrm H_2(G)$ for its Schur multiplier, and write $n^{et} \subset n$ for the subgroup coming from Lie theory. I could be wrong, but I think that $n^{et}$ has order coprime to $p$, and that $n^{exc} = n / n^{et}$ has order a power of $p$, so that $n$ splits as $n^{et} \times n^{exc}$. (The labels are "etale" and "exceptional".) If so, then there is a well-defined "purely-exceptional central extension" which is the cover $n^{exc}\cdot G$. One could ask if this group is a subsporadic? For example, $3A_7 \subset \mathrm{Co}_1$. Note that, by throwing away the unexceptional double cover of $A_7$, I don't need to use $2\mathrm{Co}_1$, and I could ask if that always happens: are purely-exceptional central extensions always subgroups of simple sporadic groups?
(Note that some groups of Lie type are groups of Lie type in more than one way, with different defining characteristics. My impression is that all such cases are furthermore subgroups of sporadic groups. For example, the exceptional isomorphism $A_8 = \mathrm{GL}_4(\mathbb{F}_2)$ arises, and is best explained inside, $M_{24}$.)
