For $n \ge 8$ the Schur multiplier $H_2(BA_n, \mathbb{Z})$ (where $A_n$ denotes the alternating group) stabilizes to $\mathbb{Z}_2$, and hence there is a universal central extension $\widetilde{A}_n$ of $A_n$ by $\mathbb{Z}_2$.

Question 1:Do any interesting groups (e.g. central extensions of sporadic simple groups) occur as centralizers (of elements) in $\widetilde{A}_n$?

Question 2:What are their Schur multipliers?

**Motivation** (feel free to ignore): Write $X = B \widetilde{A}_n$. In trying to find a relatively concrete answer to this question, I was led to a map

$$H_3(X, \mathbb{Z}) \to \pi_3(\mathbb{S}) \cong \mathbb{Z}_{24}$$

which I believe is an isomorphism for sufficiently large $n$. In any case, this map, regarded as an element of $H^3(X, \mathbb{Z}_{24})$, transgresses to a class in

$$H^2(LX, \mathbb{Z}_{24})$$

where for a group $G$, $LBG$ is the (classifying space of the) adjoint quotient $G/G$, the groupoid whose objects are elements of $g$ and whose morphisms come from conjugation. In particular, this transgressed class restricts to a distinguished family of classes in

$$H^2(C(g), \mathbb{Z}_{24})$$

where $C(g)$ denotes the centralizer and $g \in \widetilde{A}_n$. This gives a distinguished family of central extensions of the groups $C(g)$ by $\mathbb{Z}_{24}$, and it seems interesting to ask what these central extensions are. In particular (feel extremely free to ignore) the Schur multipliers of the sporadic simple groups are all subgroups of $\mathbb{Z}_{24}$, and it would be nice if this construction in some sense explained that....