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My question is, do we know anything description the automorphism group of free quandles with relative smaller number of generating set $S$? For example, for each $2\leq |S|\leq 4$, is $\mathrm{Aut}(FQ(S))$ isomorphic to any familiar group?
Yes, it's a known group, known as group of symmetric automorphisms of the free group.
The free quandle $\mathrm{FQ}_S$ can be realized as the set of conjugates of generators in the free group $\mathrm{F}_S$, with law $a\lhd b=aba^{-1}$, $b\rhd a=a^{-1}ba$. In particular, the action by conjugation of $F_S$ induces an action by quandle automorphisms, which for $|S|\ge 2$ is faithful and extends to a faithful action of $\mathrm{F}_S\rtimes\mathfrak{S}(S)$, the latter permuting generators.
For $|S|=2$ I don't see right now if this is the all automorphism group. For $|S|\ge 3$ it is not: indeed letting $a,b$ be the first two free generators, we can map $a$ to $a\lhd b$, and fix all other generators, and this is not induced by conjugation.
Every quandle $Q$ has a universal enveloping group. This is an initial object in the category of quandle homomorphisms $Q\to G$ into groups, $G$ being a quandle under $g\lhd h=ghg^{-1}$.
For the free quandle $\mathrm{FQ}_S$ the universal enveloping group is just the free group $\mathrm{F}_S$. This is obvious: the less obvious, but important fact is that $\mathrm{FQ}_S\to\mathrm{F}_S$ is injective. Hence we deduce a canonical homomorphism $\mathrm{Aut}_{\mathrm{Qua}}(\mathrm{FQ}_S)\to\mathrm{Aut}_{\mathrm{Grp}}(F_S)$, which is therefore injective too. Hence, $\mathrm{Aut}_{\mathrm{Qua}}(\mathrm{FQ}_S)$ is the set of automorphisms of the group $F_S$ preserving $\mathrm{FQ}_S$, i.e., preserving the subset defined as union of conjugacy classes of free generators.
This group is known as symmetric automorphism group of the free group $F_S$. It splits as the semidirect product $P\Sigma_S\rtimes\mathfrak{S}(S)$, where $\mathfrak{S}$ permutes generators, and $P\Sigma_S$ is the group of group automorphisms of $F_S$ mapping each generator to a conjugate. The latter has been extensively studied for $S$ finite. For $|S|$ finite it's finitely presented (McCool [1]). It has the name of groups of basis-conjugating automorphisms, or group, or group of pure symmetric automorphisms.
For $n=2$ $P\Sigma_2$ is reduced to inner automorphisms of $F_2$, but $P\Sigma_S$ it is larger for $|S|\ge 3$: letting $a,b$ be the first two free generators, we can map $a$ to $a\lhd b$, and fix all other generators, and this is not induced by a conjugation.
[1] McCool, J. On basis-conjugating automorphisms of free groups. Canad. J. Math. 38 (1986), no. 6, 1525-1529.