An extended local Lorentz symmetry in four-dimensional (4D) theory is considered. A source of this symmetry is a group of general linear transformations of four-component Majorana spinors $GL(4,M)$ which is isomorphic to $GL(4,R)$ and is the covering of an extended Lorentz group in a 6D Minkowski space $M(3,3)$ including superluminal and scaling transformations. Physical space-time is assumed to be a 4D pseudo-Riemannian manifold. To connect the extended Lorentz symmetry in the $M(3,3)$ space with the physical space-time, a fibre bundle over the 4D manifold is introduced with $M(3,3)$ as a typical fibre. The action is constructed which is invariant with respect to both general 4D coordinate and local $GL(4,M)$ spinor transformations. The components of the metric on the 6D fibre are expressed in terms of the 4D pseudo-Riemannian effective metric and two extra complex fields: 4D vector and scalar ones. These extra fields describe in the general case massive particles interacting with an extra $U(1)$ gauge field