Let $n \ge 4$ be a natural number.
Consider the quotient map from the double cover $2 \cdot A_n$ of the alternating group $A_n$ to $A_n$. For any conjugacy class in $A_n$ of size $r$, the inverse image in $2 \cdot A_n$ has size $2r$. This inverse image is either a single conjugacy class or "splits" as a union of two conjugacy classes each of size $r$. Which case occcurs depends on the original conjugacy class we chose.
I want a combinatorial criterion (see Note 2) on the cycle type of the conjugacy class in $A_n$ that helps determine which of these cases occur (actually, there may be one or two conjugacy classes in $A_n$ of a given cycle type, but even if there are two, they will behave the same way).
I'm also interested in the corresponding question for the double cover of $S_n$. Note that the double cover is not unique (there are two possibilities for each $S_n$). However, the behavior for fibers over conjugacy classes that I'm interested in should not depend on the choice of double cover (if my understanding is correct -- I haven't worked out a formal proof).
Note 1: I'm including the case of $A_n$ for $n = 4$ just so people know what I am talking about:
$A_4$ has conjugacy classes of sizes 1,3,4,4 (the two conjugacy classes of size 4 fuse in $S_4$ and correspond to cycle type 3 + 1, whereas the conjugacy class of size 3 corresponds to cycle type 2 + 2, with representative permutation $(1,2)(3,4)$).
Its double cover is isomorphic to $SL(2,3)$ and the quotient map can be viewed as a quotient map $SL(2,3) \to PSL(2,3)$. The fibers over the conjugacy classes of sizes 1,4,4, each split in two conjugacy classes. The fiber over the conjugacy class of size 3 remains a single conjugacy class of size 6, i.e., the fiber over this conjugacy class does not split.
Note 2: By "combinatorial criterion" I mean a criterion like the criterion we have for a conjugacy class of even permutations in $S_n$ to split in $A_n$: it splits if there is any cycle of even length or two cycles of equal odd length.
So, I want a criterion that looks at the cycle type as an unordered integer partition, then uses the combinatorial and number-theoretic structure of that partition to determine splitting in the double cover.