Finite groups with bicyclic Sylow subgroups in Fitting factors

Estimates of the derived length, nilpotent length, and $p$-length are obtained for a finite solvable group $G$ in which Sylow subgroups in factors of the chain $\Phi(G)=G_0\subset G_1\subset\ldots\subset G_{m-1}\subset G_m=F(G)$ of subgroups normal in $G$ are bicyclic, i.e., are factorized by two cyclic subgroups. Here, $\Phi(G)$ is the Frattini subgroup of $G$ and $F(G)$ is the Fitting subgroup of $G$. In particular, the derived length of $G/\Phi(G)$ is at most 5, the nilpotent length of $G$ is at most 4, and the $p$-length of $G$ is at most 2 for every prime $p$.