  In this paper we consider a generally covariant theory of gravity, and extend
the generalized off-shell ADT current such that it becomes conserved for field
dependent (asymptotically) Killing vector field. Then we define the extended
off-shell ADT current and the extended off-shell ADT charge. Consequently, we
define the conserved charge perturbation by integrating from the extended
off-shell ADT charge over a spacelike codimension two surface. Eventually, we
use the presented formalism to find the conserved charge perturbation of an
asymptotically flat spacetime. The conserved charge perturbation we obtain is
exactly matched with the result of the paper \cite{6'}. These charges are as
representations of the $BMS_4$ symmetry algebra. Also, we find that the near
horizon conserved charges of a non-extremal black hole with extended symmetries
are the Noether charges. For this case our result is also exactly matched with
that of the paper \cite{15}.
