Our motivation comes from the problem of classification of discrete groups acting on Riemann surfaces of given genus $g$. If $g > 1$, then by Riemann-Hurwitz bound there are finitely many finite groups. Our aim is to classify them for small genera. A complete classification is known up to genus $4$. In case of planar actions we were able to derive an algorithm [NKS][1] which emploing algebraic invariants of the actions. In all this research understanding different approaches to equivalence of actions is extremely important. Part of the problem is to understand under which circumstances one can derive reasonable characterisation of equivalence of group actions on more general spaces. In particular, we are interesting wheather the following statement holds true:
Let $S_i$ be surfaces, $i=1,2$, and $G_i$ be finite groups of homeomorphisms acting on $S_i$. Let $p_i$ be the natural projections (regular branched coverings) $S_i\to S_i/G_i$.
Is it true that the following statements are equivalent:
(1) the actions $(G_i,S_i)$ are isomorphic,
(2) there exist homeomorphisms $\tilde f\colon S_1\to S_2$ and $f\colon S_1/G_1\to S_2/G_2$ such that $f p_1=p_2\tilde f$,
(3) there exists a homeomorphism $f\colon S_1/G_1\to S_2/G_2$ such that the induced isomorphism $f^*$ of fundamental groups takes a subgroup associated with $p_1$ onto a conjugate of a subgroup associated with $p_2$.
I have looked several classical books of Algebraic topology including those by Massey, Hatcher and monography Fundamental groups and covering spaces by Elon Lages Lima. But maximum what I found is from Armstrong: Lifting group actions to covering spaces in book Macbeath: Dicrete groups and Geometry:
The action of group $G$ on path connected and locally path connected space $X$ lifts to an action of $G$ on $\tilde{X}_H$ if and only if there is a path connected and locally path connected space Z, an action of G on Z, and based equivariant map $f \colon (Z, {q}) \to (X, {p})$ such that:
(i) $H$ is $(f,G)$ invariant, and
(ii) $f_*(\pi_1(Z,q))\subset H $
Any references to textbooks or articles are welcome. Thanks for your advises.
Citations:
Massey, William S., Algebraic topology: An introduction. 5th corr. printing, Graduate Texts in Mathematics, Vol. 56. New York Heidelberg Berlin: Springer-Verlag. XXI, 261 p. DM 49.50, $ 23.60 (1981). ZBL0457.55001.
Hatcher, Allen, Algebraic topology, Cambridge: Cambridge University Press (ISBN 0-521-79540-0/pbk). xii, 544 p. (2002). ZBL1044.55001. [1]: https://arxiv.org/abs/2203.05812
Lima, Elon Lages, Fundamental groups and covering spaces. Transl. from the Spanish by Jonas Gomes, Natick, MA: A K Peters. ix, 210 p. (2003). ZBL1029.55001.
Armstrong, M. A., Lifting group actions to covering spaces, Discrete groups and geometry, Proc. Conf., Birmingham/UK 1991, Lond. Math. Soc. Lect. Note Ser. 173, 10-15 (1992). ZBL0767.57003.