Conjugacy and the Class Equation · Superchallenge problem

In $S_6$, let $\sigma=(1\,2\,3)(4\,5\,6)$ (cycle type $(3,3)$, an EVEN permutation). Does $\sigma$'s $S_6$-conjugacy class split into two $A_6$-classes, or stay as one? Give the resulting $A_6$-class size.