On abelian group actions with TNI-centralizers

A subgroup $H$ of a group $G$ is said to be a TNI-subgroup if $N_{G}(H)\cap H^g=1$forany$g\in G,\backslash ,N_{G}(H).$Let$A$ be an abelian group acting coprimely on the finite group $G$ by automorphisms in such a way that $C_G(A)=\{g\in G:g^a=g$, for all $a\in A\}$ is a solvable TNI-subgroup of $G$. We prove that $G$ is a solvable group with Fitting length $h(G)$ is at most $h(C_G(A))+\mathrm{\ell }(A)$. In particular $h(G)\le \mathrm{\ell }(A)+3$ whenever $C_G(A)$ is nonnormal. Here, $h(G)$ is the Fitting length of $G$ and $\mathrm{\ell }(A)$ is the number of primes dividing $A$ counted with multiplicities.