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We study authomorphisms of $Ind$-groups of polynomial automorphisms (wich aresingular) via tame approximations. Such objects were pioneeered in research byB.I.Plotkin  We obtain a number of properties of $Aut(Aut(A))$, where $A$ is thepolynomial or free associative algebra over the base field $K$. We prove thatall $Ind$-scheme automorphisms of $Aut(K[x_1,\dots,x_n])$ are inner for $n\ge3$, and all $Ind$-scheme automorphisms of $Aut(K\langle x_1,\dots, x_n\rangle)$are semi-inner.  As an application, we prove that $Aut(K[x_1,\dots,x_n])$ cannot be embeddedinto $Aut(K\langle x_1,\dots,x_n\rangle)$ by the natural abelianization. Inother words, the {\it Automorphism Group Lifting Problem} has a negativesolution. We explore close connection between the above results and theJacobian conjecture type questions, formulate the Jacobian conjecture forfields of any characteristic.

international journal of algebra and computation

On the tame authomorphism approximation, augmentation Topology of  Automorphism Groups and $Ind$-schemes, and authomorphisms of tame  automorphism groups