the question is the given above, specially in the case infinite:
If the action of $G$ is transitive, then there is a bijection between the fixed points of the stabilizer of a element $a$ and the lateral classes of the stabilizer of $a$ in the normalizer of the stabilizer of $a$?.
My bijection is $G_{ah}$(this denotes a lateral class of the stabilizer of a) to $ha$ where $a$ is the point of the stabilizer, this function is well-defined and is one to one. Since $G$ acts transitively then all point $y$ is equal to a $ga$ with $g \in G$, and if this is in the fixed points then the $g$ is such that the conjugate of $G_a$ is contained in $G_a$. In orden to prove that the function is a surjection, i need that the $g$ given above normalizes $G_a$, and this is precisely the problem. I can assure something about the $g$?
Thank you for the further answers
Addendum OP made me realize that in the finite case there is a fine point here. So I believe that in the general case the bijection asked for does not involve the normalizer $$ N_{G}(G_{\alpha}) = \{ g \in G : g G_{\alpha} g^{-1} = G_{\alpha} \}, $$ but the related object $$ \tilde{N}_{G}(G_{\alpha}) = \{ g \in G : g G_{\alpha} g^{-1} \subseteq G_{\alpha} \}. $$ Note that $\tilde{N}_{G}(G_{\alpha})$ is a submonoid of $G$, but need not be a subgroup of $G$. (I will provide an example later.) It coincides with the normalizer, and thus it is a subgroup, if $G$ is finite, or a torsion group.
Let $G$ act transitively on a set $A$, let $a \in A$, and let $B$ the set of points fixed by $G_{a}$.
Let $b \in B$. Then there exists $g$ such that $a g = b$. Let $h \in G_{a}$. Then $a g h = b h = b = a g$, so that $g h g^{-1} \in G_{a}$. In other words, $g \in \tilde{N}_{G}(G_{a})$.
Conversely, if $g \in \tilde{N}_{G}(G_{a})$, then $b = a g$ is fixed by $G_{a}$.
Now we are ready to set the correspondence. Start with the map $$ \tilde{N}_{G}(G_{a}) \to B $$ given by $g \mapsto a g$. We have shown that this is surjective.
Now $a g = a g'$ if and only if $g' g^{-1} \in G_{a}$, so this yields a bijection as requested between the cosets of $G_{a}$ in $\tilde{N}_{G}(G_{a})$ and $B$.
Addendum: an example Consider the additive abelian group $$A = \Bbb{Z}\left[ \dfrac{1}{2}\right] = \left\{ \dfrac{z}{2^{n}} : z \in \Bbb{Z}, n \ge 0 \right\}.$$ Let $c$ be the automorphism of $A$ given by $a \mapsto 2 a$, and let $G = A \langle c \rangle$ be the natural semidirect product.
Let $H = \Bbb{Z} \le A$, and consider the action of $G$ on the cosets of $H$. Let $\alpha = H$. Then $G_{\alpha} = H$. Now $N_{G}(G_{\alpha}) = A$, whereas $\tilde{N}_{G}(G_{\alpha})$ is the submonoid generated by $A$ and $c$, that is $$\tilde{N}_{G}(G_{\alpha}) = \{ a c^{n} : a \in A, n \ge 0 \}.$$