Almost nilpotency of an associative algebra with an almost nilpotent fixed-point subalgebra

Let A be an associative algebra of arbitrary dimension over a field F and G a finite group of automorphisms of A of order n, prime to the characteristic of F. Denote by AG={a∈A|ag=afor  allg∈G} the fixed-point subalgebra. By the classical Bergman-Isaacs theorem, if AG is nilpotent of index d, i.e. (AG)d=0, then A is also nilpotent and its nilpotency index is bounded by a function depending only on n and d. We prove, under the additional assumption of solubility of G, that if AG contains a two-sided nilpotent ideal I◁AG of nilpotency index d and of finite codimension m in AG, then A contains a nilpotent two-sided ideal H◁A of nilpotency index bounded by a function of n and d and of finite codimension bounded by a function of m, n and d. An even stronger result is provided for graded associative algebras: if G is a finite (not necessarily soluble) group of order n and A=⨁g∈GAg is a G-graded associative algebra over a field F, i.e. AgAh⊂Agh, such that the identity component Ae has a two-sided nilpotent ideal Ie◁AG of nilpotency index d and of finite codimension m in Ae, then A has a homogeneous nilpotent two-sided ideal H◁A of nilpotency index bounded by a function of n and d and of finite codimension bounded by a function of n, d and m.