  In this paper, we provide some generalization of the concept of fusion frames
following that evaluate their representability via a linear operator in Hilbert
$C*$-module. We assume that $\Upsilon _\xi$ is self-adjoint and $\Upsilon
_\xi(\frak{N} _\xi)= \frak{N} _\xi$ for all $\xi \in \mathfrak{S}$, and show
that if a
  $g-$fusion frame $\{(\frak{N} _\xi, \Upsilon _\xi)\}_{\xi \in \mathfrak{S}}$
is represented via a linear operator $\mathcal{T}$ on
  $\hbox{span} \{\frak{N} _\xi\}_{ \xi \in \mathfrak{S}}$, then $\mathcal{T}$
is bounded. Moreover, if $\{(\frak{N} _\xi, \Upsilon _\xi)\}_{\xi \in
\mathfrak{S}}$ is a tight $g-$fusion frame, then $\Upsilon_\xi $ is not
represented via an invertible linear operator on $\hbox{span}\{\frak{N}
_\xi\}_{\xi \in \mathfrak{S}}$, We show that, under certain conditions, a
linear operator may also be used to express the perturbation of representable
fusion frames. Finally, we'll investigate the stability of this fusion frame
type.
