at.algebraic topology - Involutions of $S^2$

are there some complete results on the involutions of 2 sphere?

at least I have three involutions:
(let $mathbb{Z}_2={1,g}$,and $S^2={(x,y,z)inmathbb{R}^3|x^2+y^2+z^2=1}$)

1.$g(x,y,z)=(-x,-y,-z)$(antipodal map) with null fixed point set,and orbit space $mathbb{R}P^2$
actully,for free involution on $S^n$ with $nleq3$,the orbit space is homeomorphic to real projective space (Livesay 1960)

2.$g(x,y,z)=(-x,-y,z)$ (rotation $pi$ rad around $z$ axis) with fixed point set $S^0$(the north pole and south pole) and orbit space $S^2$.

3.$g(x,y,z)=(x,y,-z)$(reflection along $z=0$) with fixed point set $S^1$ (the equator)and orbit space $D^2$

i want to know if there are some other involutions over 2-sphere.
here we take two involutions as equivalent if there are conjugate in the homeomorphism group of $S^2$

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