  We present a theoretical investigation of the resonant elastic scattering of
twisted light, carrying angular momentum, by partially stripped ions. Special
emphasis is placed on a question of whether the scattered radiation is also
twisted. In order to investigate such an "angular momentum transfer", we
develop an approach that allows us to find a quantum state of the final photon
without projecting it onto a detector state. A general expression for this
so--called \textit{evolved} state of outgoing radiation is derived and it can
be used to analyze the resonant scattering by any ion, independently of its
shell structure. Here, we illustrate our approach with the strong electric
dipole $n S_{0} \to n' P_{1} \to n S_{0}$ transitions, which play an important
role for the Gamma Factory project at CERN. For the incident Bessel light, the
scattered radiation is shown to be in a superposition of twisted modes with the
projections of the total angular momentum $m_f = 0, \pm 1$. The larger values
of $m_f$ can be efficiently generated by inducing transitions of higher
multiplicity. This angular momentum transfer, together with a remarkable cross
section that is many orders of magnitude larger than that of the backward
Compton scattering, makes the resonant photon scattering an effective tool for
the production of twisted x-- and even gamma--rays at the Gamma Factory
facility.
