Splitting criterion for conjugacy classes in double cover of alternating group

Goal of the statement
The goal of the statement is as follows. Consider a natural number $$n \ge 4$$. We are interested in the quotient map from the double cover of alternating group $$2 \cdot A_n$$ to the alternating group $$A_n$$.

For any conjugacy class $$c$$ of $$A_n$$ of size $$r$$, the inverse image of $$c$$ in $$2 \cdot A_n$$ has size $$2r$$. This inverse image is either a single conjugacy class of size $$2r$$ or it splits as a union of two conjugacy classes, each of size $$r$$. The goal is to determine a condition on the cycle type of the permutations in the conjugacy class that controls whether or not the inverse image splits.

Explicit condition
Consider a conjugacy class $$c$$ of $$A_n$$ of size $$r$$ and element order $$d$$ whose cycle type is a given unordered integer partition of $$n$$. Note that any conjugacy class in $$A_n$$ has a unique cycle type. Conversely, for every unordered integer partition with an even number of cycles of even size, there is either one or two conjugacy classes with that cycle type (more on this later). The explicit condition on the unordered integer partition is as follows:

Condition rewritten in terms of number of conjugacy classes for a given cycle type
Consider an unordered integer partition. If the unordered integer partition has an even number of even parts, then it is the cycle type of an even permutation. For even permutations, we would like to know, first, how many conjugacy classes there are in $$A_n$$ with that cycle type. This number is either 1 or 2, depending on whether the conjugacy class splits from $$S_n$$ to $$A_n$$. For each such conjugacy class, we would like to know whether its inverse image in $$2 \cdot A_n$$ splits.

To answer both these questions, we must combined this splitting criterion with the splitting criterion for conjugacy classes in the alternating group. The overall conclusions are below. REMEMBER THAT EVERYTHING BELOW IS FOR PARTITIONS WITH AN EVEN NUMBER OF EVEN PARTS, i.e., those that correspond to even permutations. Note that the first two columns are hypotheses we make about the partition and the third and fourth columns are conclusions we draw about splitting:

Orders of elements
For a given unordered integer partition, the lcm of the parts gives the order of any element of $$S_n$$ with that partition as its cycle type. If the partition has an even number of even parts, the conjugacy class is in $$A_n$$ and the order as an element of $$A_n$$ is the same: the lcm of the parts.

When we take the inverse image in $$2 \cdot A_n$$, the order of elements in the inverse image is either the same or twice the order of the element in $$A_n$$. We provide below a slight modification of the previous table that includes order information:

Conclusion for number of conjugacy classes in double cover of alternating group
For more on this, see element structure of double cover of alternating group.