How to find the number of $C^0$ conjugacy classes of a $C^1$ flow in $\mathbb R$ having $n$ number of fixed points?
It is easy to show that when $n=1$ we have $3$ $C^0$ conjugacy classes and the phase portraits are just look like this 
Here in the first phase portrait the fixed point is semistable and in the second phase portrait the fixed point is sink and for the last case the fixed point is source.
Now for the case when $n=2$ we have four $C^0$ conjugacy classes and the phase portraits are look like this
In the first phase portrait the first fixed point is sink and other one is source, in the next phase portrait first one is sink and other one is semistable, for the third phase portrait one is semistable and other one is also semistable and for the last phase portrait we have first one is source and other one is semistable.
And it is also a fact that mirror image of a phase portrait belongs in the same class.
Now my question is for $n$ larger than $2$ and how to proceed for a general $n$. A hint is given :- Use group action and take the group $ \mathbb Z_2$ and calculate the number of orbits by Burnside lemma. Now I really don’t have any idea how to proceed and any kind of solution or hint is very helpful for me. Thanks in advance.