On automorphisms of Lie algebra of symmetric polynomials

Let $L_{n}$ be the free Lie algebra of rank $n$ over a field $K$ characteristic zero, $L_{n,c}=L_{n}/(L_{n}''+\gamma_{c+1}(L_{n}))$ metabelian nilpotent class $c$ algebra, and $F_{n}=L_{n}/L_{n}''$ generated by $x_1,\ldots,x_n$ zero. We call polynomial $p(X_n)$ in these algebras {\it symmetric} if $p(x_1,\ldots,x_n)=p(x_{\pi(1)},\ldots,x_{\pi(n)})$ for each element symmetric group $S_n$. The sets $L_n^{S_n}$, $F_n^{S_n}$, $L_{n,c}^{S_n}$ polynomials coincides with invariants $S_n$ $L_{n}$, $F_{n}$, $L_{n,c}$, respectively. determine groups $\text{\rm Inn}(L_{n,c}^{S_n})\cap \text{\rm Inn}(L_{n,c})$ Inn}(F_{n}^{S_n})\cap Inn}(F_{n})$ inner automorphisms $F_{n}^{S_n}$ Inn}(F_{n})$, In particular, we obtain descriptions Aut}(L_{2}^{S_2})\cap Aut}(L_{2})$ Aut}(F_{2}^{S_2})\cap Aut}(F_{2})$ $L_{2}^{S_2}$ $F_{2}^{S_2}$ Aut}(F_{2})$,