It is known that two permutations in $S_n$ are conjugate if and only if they have same cycle decomposition shape.
Given $\sigma\in A_n$, it is also well known whether conjugacy class of $\sigma$ in $A_n$ is same as that in $S_n$; this can be done just looking at cycle structure of $\sigma$.
Q. Given $\sigma$ and $\tau$ in $A_n$ and have same cycle structure. How do we decide whether $\sigma$ and $\tau$ are conjugate in $A_n$?
For example, $(123)$ and $(132)$; in $A_4$ they are not conjugate but in $A_5$ they are!
$(12345)$ and $(21345)$ are not conjugate in $A_5$ but in bigger groups they become conjugate. In general, what is the criteria to decide whether given two permutations of $A_n$ with same cycle structure are conjugate in $A_n$ or not?
Edit: (With Derek's comment):
Hypothesis 1. Given $\sigma,\tau \in A_n$. Suppose, they have same cycle structure.
Hypothesis 2. $\sigma=\sigma_1 \circ \cdots \sigma_2$ be complete decomposition (i.e. including single cycles) such that all $\sigma_i$'s are cycles of odd length (possible singleton) and distinct.
(1) We know by hyp.2 that $\sigma^{S_n}$ (conj. class of $\sigma$ in $S_n$) is union of two conj. classes in $A_n$, say $C_1,C_2$.
(2) Since $\sigma$ and $\tau$ have same cycle structure, conclusion (3) holds for $\tau$ also.
(3) By we write $\sigma^{S_n}=C_1\cup C_2$; by hypothesis 1, $\tau^{S_n}=C_1\cup C_2$
(6) My question is now, how can we decide whether $\sigma$ and $\tau$ lie in same $C_i$'s or distinct $C_i$'s?