  We study the response of an isolated quantum system governed by the
Hamiltonian drawn from the Gaussian Rosenzweig-Porter random matrix ensemble to
a perturbation controlled by a small parameter. We focus on the density of
states, local density of states and the eigenfunction amplitude overlap
correlation functions which are calculated exactly using the mapping to the
supersymmetric nonlinear sigma model. We show that the susceptibility of
eigenfunction fidelity to the parameter of perturbation can be expressed in
terms of these correlation functions and is strongly peaked at the localization
transition: It is independent of the effective disorder strength in the ergodic
phase, grows exponentially with increasing disorder in the fractal phase and
decreases exponentially in the localized phase. As a function of the matrix
size, the fidelity susceptibility remains constant in the ergodic phase and
increases in the fractal and in the localized phases at modestly strong
disorder. We show that there is a critical disorder strength inside the
insulating phase such that for disorder stronger than the critical the fidelity
susceptibility decreases with increasing the system size. The overall behavior
is very similar to the one observed numerically in a recent work by Sels and
Polkovnikov [Phys. Rev. E 104, 054105 (2021)] for the normalized fidelity
susceptibility in a disordered XXZ spin chain.
