at.algebraic topology - Group action, Fixed point set and Orbit Space

First, you might want to identify actions that are not conjugated in $mathrm{Homeo}(M)$. For example, let $mathbb{Z}timesmathbb{Z}$ act on $M$ in the following two ways: the first factor acts by a map $f:Mto M$ and the second acts trivially, or the converse. Then the actions are the same only up to an automorphism of the group.

Second, it is not clear to me what you mean by "the same orbit space". Rational rotations of the circle all have homeomorphic orbit spaces, but of course they are not conjugate in any way.

I guess that even in favorable cases, you need the (conjugacy class of the) stabilizer of each orbit to identify the action.

Last, a remark that is not directly linked to your question, but that I like to advertise: there exist (infinite families) of analytic group actions on analytic manifolds that are topologically but not $C^1$ conjugate.