  A subgroup $H$ of a group $G$ is said to be a TNI-subgroup if $N_{G}(H)\cap
H^g=1$ for any $g\in G\,\backslash \,N_{G}(H).$ 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 $h(C_G(A))+\ell(A)$. In particular $h(G)\leq \ell(A)+3$ whenever $C_G(A)$
is nonnormal. Here, $h(G)$ is the Fitting length of $G$ and $\ell(A)$ is the
number of primes dividing $A$ counted with multiplicities.
