If the conjugacy class is preserved under product.

Question

Let $ G $ is a finite group. Assume that $ a_1,a_2,b_1,b_2\in G $ such that $ a_1=ga_2g^{-1} $ and $ b_1=hb_2h^{-1} $ for $ g,h\in G $. I want to ask if $ a_1b_1 $ and $ a_2b_2 $ are in the same conjugacy class, i.e. there exists $ c\in G $ such that $ a_1b_1=ca_2b_2c^{-1} $.

I find that this problem can be reduced to the case that $ b_1=b_2 $. However I do not know how to go on, can you give me some hints or references?

Answer 1

Counterexample in $S_4$: $a_1=(12), a_2=(13)$ and $b_1=(1234), b_2=(1432)$: $a_1b_1=(234), a_2b_2=(14)(23)$, and these elements have different cycle type hence not conjugate.