  A subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is M\"obius-like if every
element is conjugate to an element of $\mathrm{PSL}(2,\mathbb{R})$. In general,
a M\"obius-like subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is not necessarily
(semi-)conjugate to a subgroup of $\mathrm{PSL}(2,\mathbb{R})$, as discovered
by N. Kova\v{c}evi\'{c} [Trans. Amer. Math. Soc. 351 (1999), 4823-4835]. Here
we determine simple dynamical criteria for the existence of such a
(semi-)conjugacy. We show that M\"obius-like subgroups of
$\mathrm{Homeo}_+(\mathbb{S}^1)$ which are elementary (namely, preserving a
Borel probability measure), are semi-conjugate to subgroups of
$\mathrm{PSL}(2,\mathbb{R})$. On the other hand, we provide an example of
elementary subgroup of $\mathrm{Diff}^\infty_+(\mathbb{S}^1)$ satisfying that
every non-trivial element fixes at most 2 points, which is not isomorphic to
any subgroup of $\mathrm{PSL}(2,\mathbb{R})$. Finally, we show that
non-elementary, non-locally discrete subgroups acting with at most $N$ fixed
points are conjugate to a dense subgroup of some finite central extension of
$\mathrm{PSL}(2,\mathbb{R})$.
